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Van  Antwerp,  Pragg  &  Co. 

Cincinnati  &  New  York. 


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WHITE'S   GRADED-SCHOOL   SERIES. 


AN 


INTERMEDIATE 


ARITHMETIC, 


MENTAL  AND  WRITTEN  EXERCISES 


NATURAL  SYSTEM  OF  INSTRUCTION. 


By  E.  E.  white,  M.A. 


VAN  ANTWERP,  BRAGG  A  CO., 
CINCINNATI.  NEW  YORK. 


Entered  according  to  Act  of  Congress,  in  the  year  1870,  by 

WILSON,   HINKLE  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the 


Southern  District  of  Ohio. 


Entered  according  to  Act  of  Congress,  in  the  year  1873,  by 

WILSON,  HINKLE  &  CO., 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington,  D.  C. 


Copyright,  1876,  by  Wilson,  Hinkle  &  Co. 


EDUCATION  DEPT, 


ECLECTIC    press: 

VAN  ANTWERP,  BRAGG  k  CO., 

CINCINNATI. 


l^i^ 


PREFACE. 


It  is  claimed  for  this  treatise  that  it  possesses  three  very 
important  characteristics,  to  wit: 

1.  It  is  specially  adapted  to  the  grade  of  pupils  for  which  it  is 
designed.  It  presents  only  those  operations  and  principles  which 
can  be  mastered  by  intermediate  classes,  and  each  subject  is 
treated  as  thoroughly  as  the  advancement  of  such  pupils  will 
permit.  It  is  also  believed  that  the  subjects  are  introduced  in 
the  best  possible  order.  There  are  reasons  in  favor  of  placing 
United  States  Money  before  Fractions,  but  stronger  reasons 
favor  the  arrangement  in   this  work. 

2.  It  combines  mental  and  ivritten  arithmetic  in  a  practical  and 
philosophical  manner.  This  is  done  by  making  every  mental  ex- 
ercise preparatory  to  a  written  one ;  and  thus  these  two  classes 
of  exercises,  which  have  been  so  unnaturally  divorced,  are 
united  as  the  essential  complements  of  each  other.  This  union 
is.  natural  and  complete;  and,  as  a  consequence,  the  several 
subjects  are  treated  in  much  less  space  than  is  possible  when 
mental  and  written  exercises  are  presented  in  separate  books. 

3.  It  faithfully  embodies  the  Inductive  Method.  Instead  of  at- 
tempting to  deduce  a  principle  or  rule  from  a  single  example, 
as  is  usually  done,  each  process  is  developed  inductively,  and 
the  succcvssive  steps  are  thoroughly  mastered  and  clearly  stated 
by  the  pupil  before  he  is  confronted  with  the  author^s  rule. 
This  method  not  only  places  "processes  before  rules,"  but  it 
teaches  "  rules  through  processes,"  thus  observing  two  important 
inductive  maxims.  .— ^-^  ji  ^a  ^-  r-'  .^ 


IV  PREFACE. 

Attention  is  also  called  to  the  use  of  visible  illustrations  (objects 
or  pictures)  in  developing  new  ideas  and  processes.  In  the  funda- 
mental rules,  this  illustrative  or  perceptive  step  is  omitted, 
since  it  is  fully  presented  in  the  Pkimary  Arithmetic.  The 
engraved  cuts  in  Fractions,  United  States  Money,  and  Denom- 
inate Numbers,  are  specially  designed  to  be  used  as  a  means 
of  developing  and  illustrating  the  subjects  considered. 

Two  other  features,  worthy  of  special  notice,  are  the  great 
variety  of  exercises^  and  their  preeminently  progi^essive  character. 
Generally,  each  lesson  contains  both  concrete  and  abstract  ex- 
amples, and  every  new  process  or  combination  is  at  once  used 
in  the  solution  of  problems  involving  mental  analysis.  This 
arrangement  avoids  the  mechanical  monotony  which  character- 
izes long  drills  on  a  single  class  of  exercises.  The  problems, 
all  of  which  are  original,  are  so  graded  that  they  present  but 
one  difficulty  at  a  time,  and  all  difficulties  in  their  natural  order. 
The  pupil's  progress  is  thus  made  easy  and  thorough. 

It  is  hoped  that  these  and  other  features  may  commend  this 
work  to  all  progressive  teachers,  and  that  it  may  prove  as  suc- 
cessful in  the  school-room  as  its  plan  is  natural  and  simple. 

Columbus,  Ohio,  May^  1870. 


The  Last  Edition. 

The  recent  addition  of  the  more  useful  processes  in  Per- 
centage and  Mensuration  adapts  the  work  to  those  pupils  who 
do  not  attend  school  long  enough  to  master  an  arithmetic  de- 
signed for  advanced  classes.  It  now  presents  a  short  course 
IN  arithmetic,  with  thorough  drills  in  all  elementary  processes, 
and  with  a  brief  and  simple  treatment  of  those  practical  appli- 
cations which  are  most  frequently  used  in  business. 

Lafayette,  Ind.,  May,  1876. 


SUGGESTIONS  TO  TEAOHEES. 


In  the  preparation  of  this  work  two  facts  were  kept  in  view, 
viz.:  (1)  that  it  is  to  be  studied  by  pupils  who  must  largely 
depend  upon  the  living  teacher  for  explanations;  and  (2)  that 
those  methods  wiiich  are  most  natural  and  simple  are  most  suc- 
cessful in  practice.  Hence,  its  paojes  are  not  cumbered  with 
long  verbal  explanations  and  peculiar  methods,  of  little  prac- 
tical use  to  pupil  or  teacher.  The  author^  has  left  something 
for  the  teacher  to  do;  and  tliat  this  may  be  done  wisely,  he 
offers  the  following  hints  and  suggestions: 

1.  Mental  Exercises. — These  exercises  should  be  made  a  thor- 
ough intellectual  drill.  They  should  be  recited  mentally,  that 
is,  without  writing  the  results;  and,  since  the  reasoning  faculty 
is  not  trained  by  logical  verbiage,  the  solutions  should  be  con- 
cise and  simple.  See  pages  23,  88,  89.  They  should  also  be 
made  introductory  to  the  Written  Exercises,  of  which  they  are 
often  a  complete  elucidation.  The  corresponding  examples  in 
the  two  classes  of  exercises  should  be  recited  together  as  well 
as  separately.  Many  of  the  written  problems  may  also  be  solved 
mentally. 

2.  Written  Exercises. — The  pupils  should  be  required  to  solve 
every  problem  of  the  assigned  lesson  on  tlie  slate  or  paper,  and 
the  solutions  should  be  brought  to  the  recitation  for  the  teacher's 
inspection  and  criticism.  From  three  to  five  minutes  will  suffice 
to  test  the  accuracy  and  neatness  of  each  pupil's  work.  The 
mental  problems  should  also  be  solved  on  the  slate  or  paper  in 
preparing  the  lesson,  and  then  recited,  not  only  mentally  as 
above  described,  but  also  as  a  written  exercise.  This  will  in- 
crease the  number  of  written  problems,  and,  at  the  Fame  time, 
it  will  secure  a  careful  preparation  of  the  entire  lesson. 

3.  Definitions  and  Principles. — These  should  be  deduced  and 
stated  by  the  pupils  under  the  guidance  of  the  teacher,  and 
usually  in  connection  with  the  solution  of  problems.  Take  for 
illustration  the  definition  of  multiplication.  The  pupil  mul- 
tiplies 304  by  5.  The  teacher  asks,  What  have  you  done?  "I 
have  multiplied  304  by  5."  T.  Do  not  use  the  word  "multi- 
plied." (If  necessary,. the  teacher  shows  what  is  meant  by  taking 
a  thing  one  or  more  times.)     "  I   have   taken   304  five   times." 

(V) 


VI  SUGGESTIONS  TO  TEACHERS. 

T.  By  what  process  have  you  taken  304  five  times?  "By  mul- 
tiplying it.^'    T.  What,  then,  is  multiplication?    ''Multiplication 

is  the  process  of  taking  one  number ."     T.  How  many  times 

is  the  number  taken  in  the  above  example?  "It  is  taken  five 
times,  or  as  many  times  as  there  are  units  in  the  multiplier." 
T.  Now  complete  your  definition.  ^^Multiplication  is  the  process 
of  taking  one  number  cts  many  times  as  there  are  units  in  another. ^^ 
These  steps  should  be  repeated  with  other  examples,  until  the 
definition  is  clearly  reached  and  accurately  stated.  It  should 
then  be  written  and  compared  with  the  author's  definition,  which 
should  be  thoroughly  memorized. 

4.  Eules. — These  should  also  be  deduced  and  stated  by  the 
pupils.  The  true  order  is  this:  1.  A  mastery  of  the  process 
without  reference  to  the  rule.  2.  The  recognition  of  the  suc- 
cessive steps  in  order,  and  the  statement  of  each.  3.  The  com- 
bination of  these  several  statements  into  a  general  statement. 
4.  A  comparison  of  this  generalization  with  the  author's  rule. 
6.  The  memorizing  af  the  latter.  Take  for  illustration  the  rule 
for  adding  fractions.  T.  What  is  the  first  step?  "Write  the 
fractions,  separating  them  by  the  plus  sign."  (Pupils  write  an 
example.)  T.  What  is  the  second  step?  "Reduce  the  fractions 
to  a  common  denominator."  2\  What  is  the  third  step?  "Add 
the  numerators  of  the  new  fractions."  T.  The  fourth  step? 
"Under  their  sum  write  the  common  denominator."  These  ques- 
tions should  be  repeated  until  the  answers  are  promptly  and  ac- 
curately given,  and  then  they  should  be  united  in  a  general 
statement.     The  first  step  may  be  omitted  in  the  rule. 

5.  Questions  for  Review. — These  are  designed  as  a  final  test 
of  the  pupil's  knowledge.  Before  they  are  reached,  the  defini- 
tions, principles,  and  rules  should  be  thoroughly  mastered,  and 
the  pupils  should  be  able  to  make  a  topical  analysis  of  them 
and  recite  each  in  order. 

6.  Fractions. — This  section  presents  only  the  elements  of  Frac- 
tions, and  these  in  the  simplest  manner.  The  subject  is  more  ex- 
haustively treated  in  the  Complete  Arithmetic.  Tlie  reduction 
of  compound  fractions  is  made  introductory  to  the  multiplication 
of  fractions,  as  the  two  processes  are  best  taught  together. 

7.  Oine  Method. — Elementary  instruction  in  arithmetic  should 
aim  to  make  the  pupil  ready  and  accurate  in  the  use  of  one 
method  for  each  operation.  This  may  not  be  the  shortest  method 
in  every  case;  but,  as  a  rule,  the  pupil  will  reach  the  result  sooner 
by  it  than  by  a  method  that  is  less  familiar.  The  attempt  of  the 
young  pupil  to  use  several  methods,  results  in  hesitation  and 
confusion. 

For  other  suggestions  see  the  Manual  of  Arithmetic. 


coNTE:^rTS. 


SECTION  I.— Notation  and  Numeration. 

PAGE 

Oral  and  Written  Exercises 9 

Definitions,  Principles,  and  Rules 16 

Koraan  Notation 20 

SECTION  II.— Addition. 

Mental  and  Written  Exercises 23 

Definitions,  Principles,  and  Rule 35 

SECTION  III.— Subtraction. 

Mental  and  Written  Exercises       ......  37 

Definitions,  Princii)les,  and  Rule  .        .        .        .        .        .45 

SECTION  IV.— Multiplication. 

Mental  and  Written  Exercises 48 

Definitions,  Principles,  and  Rule 59 

SECTION  v.— Division. 

Mental  and  Written  Exercises 63 

Definitions,  Principles,  and  Rules 74 

SECTION  VI. — Properties  of  Numbers. 

Greatest  Common  Divisor 80 

Multiple  and  Least  Common  Multiple         ....  83 

SECTION  VII.— Common  Fractions. 

The  Idea  of  a  Fraction 86 

Reduction  of  Integers  and  Mixed  Numbers  to  Fractions   .  88 

Reductions  of  Fractions 90 

Addition  of  Fractions 96 

Subtraction  of  Fractions 98 

Multiplication  of  Fractions 101 

(vii) 


VlU  CONTENTS. 

PAGE 

Division  of  Fractions 107 

Fractional  Parts 110 

SECTION  VIII. -Decimal  Fractions. 

Numeration  and  Notation 114 

Reduction  of  Decimals 119 

Addition  of  Decimals 122 

Subtraction  of  Decimals 123 

Multiplication  of  Decimals 124 

Division  of  Decimals 126 

SECTION  IX.—United  States  Money. 

Notation  and  Definitions        .......  129 

Reduction  of  United  States  Money 132 

Addition  and  Subtraction  of  United  States  Money      .        .  133 

Multiplication  and  Division  of  United  States  Money         .  135 

Bills '.'....  139 

SECTION  X. — Denominate  Numbers. 

Reduction,  Mental  and  Written 144 

Definitions,  Principles,  and  Rules          .         .        .        .         .  171 

SECTION  XI.— Compound  Numbers. 

Compound  Addition        .        . 177 

Compound  Subtraction    .         . 180 

Compound  Multiplication 183 

Compound  Division 185 

SECTION  XII.— Percentage. 

Notation  and  Definitions .190 

The  Three  Cases 191 

Profit  and  Loss 196 

Commission,  Insurance,  Taxes,  etc 198 

Simple  Interest 199 

Discount 203 

Notes,  Drafts,  and  Bonds 207 

SECTION  XIII.— Mensuration. 

Surfaces 209 

Solids 211 


INTERMEDIATE  ARITHMETIC. 


SECTION  I. 


%%^%%%%%%% 


LESSON    I. 


ORAL  EXERCISES. 


Article  1.  —  1.  Here  are  one  hundred  balls  in  ten 
rows.  How  many  balls  are  there  in  one  row?  How 
many  balls  in  two  rows?  In  three  rows?  In  five 
rows?     In  eight  rows?     In  ten  rows? 

2.  How  many  ones  in  ten?  How  many  ones  in 
two  tens?     In  five  tens?     Eight  tens?     Ten  tens? 

(9) 


10  TNTERMEDIi^TE   ARITHMETIC. 

3.  How  *  many*  tens  ih  ten?  How  many  tens  in 
twenty?  In  thirty?  Forty?  Sixty?  Seventy? 
Eighty?     One   hundred? 

Art.  2.  When  a  number  is  expressed  by  two  figures, 
the  first  or  right-hand  figure  denotes  Units^  and  the 
second  or  left-hand  figure  denotes  Tens. 

4.  Which  figure  in  25  denotes  units?  Which  de- 
notes  tens? 

5.  How  many  tens  and  units  are  there  in  37? 
In    57?    46?    33?    50?    45?    64?    88?    94?    99? 

Art.  3.  In  reading  numbers,  the  tens  and  units  are 
read  together  as  so  many  units.  Thus,  45  is  read 
forty-five  units,  or,  more  briefly,  forty -five. 

Eead  the  following  numbers,  and  give  the  number 
of  tens  and  units  in  each : 


(6) 

(7) 

(8) 

(9) 

(10) 

(11) 

1 

11 

21 

20 

14 

67 

3 

13 

23 

40 

34 

83 

5 

15 

25 

60 

55 

75 

9 

19 

29 

80 

95 

72 

Art.  4.  When  a  number  is  expressed  by  three  fig- 
ures, the  third  or  left-hand  figure  denotes  Hundreds. 

12.  Which  figure  in  245  denotes  hundreds?     Which 
figure  denotes  tens?     Which  denotes  units? 

13.  How  many   hundreds,  tens,  and   units  in  426? 
708?    340?    235?    406?    560?    666? 

Eead  the  following  numbers,  and  give  the  number 
of  hundreds,  tens,  and  units  in  each : 


(14) 

(15) 

(16) 

(17) 

(18) 

200 

240 

302 

349 

560 

500 

550 

805 

424 

703 

700 

770 

807 

825 

909 

900 

990 

804 

448 

836 

NOTATION  AND   NUMERATION.  11 

19.  What  is  the  greatest  number  that  can  be  ex- 
pressed by  one  figure?  By  two  figures?  By  three 
figures  ? 

20.  When  numbers  are  expressed  by  figures,  in 
which  place  or  order  is  the  figure  denoting  units 
written?  The  figure  denoting  tens?  The  figure  de- 
noting hundreds? 

Art.  5.  The  first  three  figures,  viz. :  units,  tens,  and 
hundreds,  constitute  the  first  or   Units'   Period. 

'WRITTEN  EXERCISES. 

1.  Write  in  words,  4,  6,  8,  13,  14,  18,  20,  24,  30, 
34. 

2.  Write  in  words,  40,  46,  60,  67,  70,  78,  80,  83, 
87,  90,  95,  99. 

Express  in  figures  the  fi^llowing  numbers : 

(3)  (4)  (5) 

Twelve;  Twenty-one;  Twenty-three; 

Sixteen ;  Thirty-two ;  Twenty-four ; 

Eighteen;  Forty-two;  Forty-seven; 

Twenty;  Sixty-five;  Sixty-five; 

Sixty ;  Eighty-five ;  Seventy-nine ; 

Eighty.  Ninety-four.  Ninety-six. 

6.  Express  in  figures  the  numbers  composed  of  three 
tens  and  four  units ;  six  tens  and  seven  units ;  seven 
tens  and  six  units;  seven  tens. 

7.  Express  in  figures  the  numbers  composed  of  six 
tens  and  eight  units;  three  tens  and  nine  units;  nine 
tens  and  no  units ;  seven  units. 

8.  Write  in  words,  100,  150,  200,  280,  300;  350, 
390,    560,    607,    803,    340,    and    908. 


12  INTERMEDIATE    ARITHMETIC. 

Express  in  figures  the  following  numbers : 

(9)  (10) 

Two  hundred  ;  Four  hundred  and  five ; 

Five  hundred  ;  Five  hundred  and  six ; 

Seven  hundred ;  Six  hundred  and  four ; 

Three  hundred  and  forty;  Four  hundred  and  forty-five; 

Six  hundred  and  seventy ;  Eight  hundred  and  thirty-seven; 

Nine  hundred  and  thirty.  Nine  hundred  and  twenty-seven. 

11.  Express  in  figures  the  numbers  composed  of 
three  hundreds,  five  tens,  and  four  units;  six  hun- 
dreds, four  tens,  and  three  units ;  five  hundreds,  seven 
tens,  and  no  units. 

12.  Express  in  figures  the  numbers  composed  of 
eight  hundreds  and  six  tens;  five  hundreds  and  four 
tens ;  seven  hundreds  and  five  units ;  two  hundreds 
and  six  units;   six  tens. 

13.  What  number  is  composed  of  3  hundreds,  0 
tens,  and  6  units?  2  hundreds  and  3  tens?  4  hun- 
dreds and  6  units?    5  hundreds  and  8  tens? 

14.  What  number  is  composed  of  5  tens  and  8  units? 
6  hundreds  and  5  units?    7  hundreds  and  6  tens? 


LESSON    II. 

2'hoiisa)ids*    "Period —  ^'hoiisands,   2en-ihousa7ids, 
Mund^^ed- thousands, 

ORAIi    EXERCISES. 

Art.  6.  When  a  number  is  expressed  by  four  figures, 
the  fourth  or  left-hand  figure  denotes  Thousands. 

1.  How  many  thousands  in  4,635?     3,045?     6,309? 
7,554?     5,384?     8,054?     5,006? 

2.  Read   the   units'  period   in  6,325;    5,080;    7,009; 
3,406;    5,800;    6,370;    7,590;    8,008. 


NOTATION  AND  NUMERATION.  13 

Eead  the  following  numbers: 

(3)  (4)  (5)  (6)  (7) 

1,000  2,200  1,020  2,007  3,432 

3,000  4,400  3,040  4,001  4,568 

5,000  6,600  5,060  5,003  5,608 

7,000  8,800  7,090  6,005  7,893 

9,000  /        9,900  9,070  8,009  9,890 

Art.  7.  When  a  number  is  expressed  by  five  fig- 
ures, the  fifth  or  left-hand  figure  denotes  tens  of 
thousands,  or  Ten-thousands. 

8.  How  many  ten-thousands  in  45,684?  50,480? 
38,305?     15,056?    80,650? 

9.  How  many  ten-thousands  and  thousands  in 
36,308?    48,500?     60,070?    85,350?     90,308? 

Art.  8.  In  reading  a  number  expressed  by  five  fig- 
ures, the  fifth  and  fourth  figures  are  read  together  as 
so  many  thousands.  Thus,  45,000  is  read  forty-five 
thousand. 

Eead  the  following  numbers : 


(10) 

(11) 

(12) 

(13) 

10,000 

21,000 

34,400 

53,333 

30,000 

44,000 

53,440 

16,089 

50,000 

63,000 

67,444 

99,008 

70,000 

84,000 

48,307 

28,045 

90,000 

99,000 

39,600 

67,909 

Art.  9.  When  a  number  is  expressed  by  six  figures, 
the  sixth  or  left-hand  figure  denotes  hundreds  of  thou- 
sands, or  Hundred-thov sands. 

14.  How  many  hundred-thousands  in  534,000? 
308,000?      650,430?      508,080? 

15.  How  many  hundred-thousands,  ten-thousands, 
and  thousands  in  354,000?  607,800?  350,307? 
193,240  ?      470,386  ? 


14  INTERMEDIATE  ARITHMETIC. 

Art.  10.  In  reading  a  number  expressed  by  six  fig- 
ures, the  sixth,  fifth,  and  fourth  figures  are  read  together 
as  thousands.  Thus,  452,000  is  read  four  hundred  and 
fifty-two  thousand. 

Eead  the  following  numbers  : 


(16) 

(17) 

(18) 

(19) 

200,000 

250,000 

845,630 

603,408 

400,000 

360,000 

803,084 

490,732 

600,000 

580,000 

760,432 

308,400 

800,000 

730,000 

900,425 

600,550 

900,000 

960,000 

807,708 

707,700 

Art.  11.  The  fourth,  fifth,  and  sixth  figures  of  a 
number  constitute  the  Thousands'  Period, 

20.  Eead  the  thousands'  period  in  the  16th,  17th, 
18th,  and  19th  examples. 

21.  How  many  orders  in  units'  period?  In  thou- 
sands' period  ? 

22.  What  are  the  names  of  the  three  orders  in 
units'  period?     In  thousands'  period? 

23.  How  may  the  two  periods  be  separated? 
Ans.   By  a  comma, 

WKITTEN    EXERCISES. 

1.  Write  in  words,  3000 ;  4060;  3580;  7086;  6606; 
and  8080. 

2.  Write  in  words,  4400 ;  5008 ;  6070 ;  8506 ;  5087  ; 
7600;  and  3003. 

3.  Express  in  figures,  three  thousand ;  seven  thou- 
sand ;  nine  thousand ;  four  thousand  five  hundred ; 
eight  thousand  nine  hundred. 

4.  Express  in  figures,  two  thousand  four  hundred 
and  forty ;  four  thousand  six  hundred  and  sixty ;  five 
thousand  eight  hundred ;  six  thousand  five  hundred 
and  twenty -five. 


NOTATION  AND   NUMERATION.  15 

5.  Express  in  figures,  seventy-five;  two  hundred  and 
forty ;  three  hundred  and  six ;  five  hundred  and  forty- 
five;  four  thousand. 

6.  Express  in  figures,  four  hundred  and  forty ;  five 
hundred  and  ninety ;  seven  thousand  eight  hundred ; 
eight  thousand  and  fi^j. 

7.  Write  in  words,  10000;  25000;  40500;  36000; 
44000;  30400;  45080;  64008;  89800. 

8.  Express  in  figures,  forty-five  thousand  five  hun- 
dred and  four;  sixty  thousand  seven  hundred  and 
ninety ;  thirty-eight  thousand  and  twenty ;  ninety-six 
thousand  and  eighty-four. 

9.  Express  in  figures,  four  hundred  and  twenty; 
seven  hundred  and  eighty-nine ;  four  thousand  and 
fifty-seven ;  seventy-five  thousand ;  sixteen  thousand 
and  ninety-eight. 

10.  Express  in  figures,  as  one  number,  87  thousand 
327  units;  60  thousand  405  units;  70  thousand  346 
units ;  4  thousand  40  units ;  5  thousand  5  units ;  95 
thousand  406  units. 

11.  Express  in  figures,  as  one  number,  88  thousand 
88  units ;  8  thousand  80  units ;  65  thousand  60  units ; 
6  thousand  600  units ;    60  thousand. 

12.  Write  in  words,  300000;  440000;  334000;  245500; 
304800;   450340. 

13.  Express  in  figures,  four  hundred  thousand  ;  six 
hundred  thousand;  eight  hundred  and  forty  thousand  ; 
seven  hundred  and  sixty  thousand. 

14.  Express  in  figures,  nine  hundred  and  fifty  thou- 
sand four  hundred ;  four  hundred  and  fifty-five  thou- 
sand two  hundred  and  eighty. 

15.  Separate  the  following  numbers  into  periods : 
3080 ;  44004 ;  400080  ;  20066 ;  109038 ;  160006 ;  809090 ; 
706030;    40004;   30030. 


16  INTERMEDIATI5   ARITHMETIC. 

LESSON     III. 

DEFINITIONS,  PEINOIPLES,  AND  EXILES. 

Art.  12.  Arithmetic  is  the  science  of  numbers,  and 
the  art  of  numerical  computation. 

A  Number  is  a  unit  or  a  collection  of  units. 
A  Unit  is  one  thing  of  any  kind. 
An  Integer  is  a  whole  number. 

Art.  13.  There  are  three  methods  of  expressing 
numbers : 

1.  B}^  words;   as,  five,  ^fty^  etc. 

2.  By  letters^  called  the  Roman  method.    (Art.  23.) 

3.  By  figures^  called  the  Arabic  method. 

Art,  14.  Notation  is  the  art  of  expressing  numbers 
by  figures  or  letters. 

Numeration  is  the  art  of  reading  numbers  ex- 
pressed by  figures  or  letters. 

The  word  Notation  is  commonly  used  to  denote  the  Arabic 
method,  which  expresses  numbers  by  figures. 

Art.  15.  In  expressing  numbers  by  figures,  ten  char- 
acters are  used,  viz. :    0,  1,  2,  3,  4,  5,  6,  7,  8,  9. 

The  first  of  these  characters,  0,  is  called  JVaught, 
or  Cipher,  It  denotes  nothing^  or  the  absence  of 
mimher. 

The  other  nine  characters  are  called  Significant 
Figures,  They  each  express  one  or  more  units. 
They  are  also  called  Digits. 

Art.  16.  The  successive  figures  which  express  a 
number,  denote  successive  Orders  of  Units.  These 
orders  are  numbered  from  the  right ;  as,  first,  second, 
third,  fourth,  fifth,  and  so  on. 


NOTATION   AND  NUMERATION.  17 

A  figure  in  units'  phice  denotes  units  of  the  first 
order;  in  tens'  place,  U7iits  of  the  second  order;  in 
hundreds'  place,  units  of  the  third  order,  and  so  on  — 
the  term  units  being  used  to  express  ones  of  any  order. 

Art.  17.  Ten  units  make  one  ten,  ten  tens  make 
one  hundred,  ten  hundreds  make  one  thousand ;  and, 
generally,  ten  units  of  any  order  7nake  one  unit  of  the 
next  higher  order. 

Note. — The  teacher  can  make  this  principle  plain  by  means 
of  the  illustration  given  on  page  9.  It  is  easily  shown  that  10 
ones  or  units  equal  1  ten,  and  that  10  tens  equal  1  hundred. 

Art.  18.  Figures  have  two  values,  called  Simple 
and  Local. 

The  Simjjle  Value  of  a  figure  is  its  value  when 
standing  in  units'  place. 

The  Local  Value  of  a  figure  is  its  value  arising 
from   the   order   in   which    it   stands. 

When  3,  for  example,  stands  alone,  or  in  the  first 
order,  it  denotes  3  units;  when  it  stands  in  the 
second  order,  as  in  34,  it  denotes  3  tens;  when  it 
stands  in  the  third  order,  as  in  354,  it  denotes  3 
hundreds.  Hence,  the  local  value  of  figures  increases 
from  right  to  left  in  a  tenfold  ratio. 

The  local  value  of  each  of  the  successive  figures  which 
express  a  number,  is  called  a  Term.  The  terms  of 
325  are  3  hundredths^  2  tens,  and  5  units. 

Art.  19.  The  figures  denoting  the  successive  orders 
of  units,  are  divided  into  groups  of  three  figures 
each,  called  Periods.  The  first  or  right-hand  period 
is  called  Units;  the  second.  Thousands;  the  third. 
Millions;  the  fourth,  Billions;  the  fifth,  Trillions; 
the  sixth,  Quadrillions;  the  seventh,  Quintillions ;  etc. 
I.  A.— 2. 


18 


INTERMEDIATE   ARITHMETIC. 


Art.  20.  The  three  orders  of  any  period,  counting 
from  the  right,  denote,  respectively,  Units ^  Tens^  and 
Hundreds,  as  shown  in  the  table : 


o 


o 


e 


K    H    P 

5    5    5 


1    1    1 


5th  Period,       4th  Period,        3d  Period,         2d  Period,       1st  Period, 
Trillions.       Billions.         Millions,       Thousands,       Units. 

The  several  orders  may  be  named  more  briefly  by 
calling  the  first  order  of  each  period  by  the  7ianie  of 
the  period,  and  omitting  the  word  "of"  after  tens 
and  hundreds,  thus : 


.2 

V      O 

■73 

S 

o 

1 

1 

1 

i 

'6 

i 

:3 

'd 

e 
a 

^ 

'§      ^ 

O 

'2 

1 

.2 

2i 

13 

1 

.2 

22 

o 

c«-       B 

"C 

H 

3 

1 

§ 

3 

i 

1 

B 

o 

H 

O 

5    5 

5 

od. 

,    4 

4 

4 

3 

3 

2 

2 

2 

,  1 

1st 

1      1 

5th  Perj 

4th  Period. 

3d  Period. 

2d  Period. 

Period. 

Art.  21.  KuLE  FOR  Notation. — Begin  at  the  left,  and 
write  the  figures  of  each  period  in  their  proper  orders, 
filling  all  vacant  orders  and  periods  with  ciphers. 

Art.  22.  EuLE  for  Numeration  — 1.  Begin  at  the 
right,  and  separate  the  number  into  periods  of  three 
figures  each. 


NOTATION   AND  NUMERATION.  19 

2.  Begin  at  the  left^  and  read  each  period  containing 
one  or  more  significant  figures  as  if  it  stood  alone, 
adding   its   name. 

Note.  —  The  name  of  the  units'  period  is  usually  omitted. 
WBITTEN   EXEKCISES. 

1.  Write  in  words,  20080406. 

Suggestion.— Separate  the  number  into  periods,  thus :  20,080,406. 
Then  write  each  period,  thus :  Twenty  million  eighty  thousand  four 
hundred  and  six. 

2.  Write  in  words,  50038456. 

3.  Write  in  words,  300607008. 

4.  Write  in  words,  40000300400. 

Suggestion.— Omit  the  third  period,  since  it  contains  no  sig- 
nificant figures,  thus:  Forty  billion  three  hundred  thousand  four 
hundred. 

5.  Write  in  words,  3450000067. 

6.  Eead  3000080040;    10080603400. 

7.  Eead  15000407030;    5075803004. 

8.  Eead  400440300500;    130030003003. 

9.  Express  in  figures,  twelve  billion  forty-six  mill- 
ion and  nine. 

Process. — First,  write  12,  with  a  comma  after  it,  to  form 
the  fourth  or  billions'  period,  thus:  12, ;  then  write  46  in  the 
next  period,  filling  the  vacant  order  with  a  cipher,  thus: 
12,046,;  then,  as  there  are  no  thousands,  fill  the  next  three 
orders  with  ciphers,  thus:  12,046,000,;  and,  finally,  write  9 
in  the  units'  period,  filling  the  vacant  orders  with  ciphers, 
thus:  12,046,000,009. 

10.  Express  in  figures,  fifty  million  thirty-two  thou- 
sand six  hundred  and  forty. 

11.  Three  hundred  million  nine  thousand  two  hun- 
dred and  six. 


20  INTERMEDIATE    ARITHMETIC. 

12.  Forty-eight    billion     seventeen     thousand     and 
sixty-four. 

13.  Five  million  five  thousand  and  five. 

14.  One  million  one  hundred  thousand  and  ten. 

15.  Three  trillion  three  hundred  million  three  hun- 
dred and  three. 

16.  Sixty-two  million  three  hundred  thousand  and 
fortj^-nine. 

17.  Five  hundred  million  five  thousand. 

18.  Four  hundred  and  six  thousand  five  hundred 
and  seven. 

19.  Two  million  ten  thousand  and  eighty. 

20.  Ninety  million  seven  thousand  four  hundred 
and    ninety. 

21.  Four  hundred  million  forty  thousand  four  hun- 
dred and  four. 

22.  Thirty  billion  seventy-five  thousand. 

23.  Nine  billion  nine  thousand  and  nine. 

24.  Fifty-four  million  eighty-seven  thousand  and 
eighty-six. 

25.  Two  hundred  and  two  thousand  five  hundred 
and  eighty. 

26.  Fifty  billion  fifty  million  ^ve  hundred  thousand 
and  seven, 

27.  Seventeen  billion  seven  hundred  thousand  three 
hundred  and  six. 

28.  Ninety  million  ten  thousand  and  fifty-five. 

LESSON    IV. 

"ROMAJV  J\rOT^TIOJ\r. 

Art.  23.  In  the  Roman  Notation,  numbers  are  ex- 
pressed by  means  of  seven  capital  letters^  viz. :  I,  Y, 
X,  L,  C,  D,  M. 


NOTATION   AND  NUMERATION. 


21 


I  stands  for  one ;  V  for  five ;  X  for  ten ;  L  for  fifty ; 
C  for  one  hundred ;  1)  for  five  hundred ;  M  for  one 
thousand. 

Art.  24.  All  other  numbers  are  expressed  by  re- 
peating or  combining  these  letters. 

1.  When  a  letter  is  repeated,  its  value  is  repeated ; 
thus :  II  represent  2 ;  XX,  20 ;  CCC,  300,  etc. 

2.  When  a  letter  is  placed  before  one  of  greater 
value,  the  less  value  is  taken  from  the  greater;  thus: 
IV  stands  for  4 ;    IX  for  9 ;    XC  for  90. 

3.  W^hen  a  letter  is  placed  after  one  of  greater 
value,  the  less  value  is  added  to  the  greater;  thus: 
VI  stands  for  6;    XI  for  11;    CX  for  110. 

Art.  25.  In  the  following  table,  numbers  are  ex- 
pressed by  letters  and  figures : 


I,    1 

VIII,    8 

XV,        15 

XL,        40 

II,      2 

IX,       9 

XVI,      16 

L,            60 

III,    3 

X,       10 

XVII,     17 

LX,        GO 

IV,    4 

XI,      11 

XVIII,  18 

LXX,     70 

V,     5 

XII,    12 

XIX,      19 

LXXX  80 

VI,    6 

XIII,  13 

XX,        20 

XC,        90 

VII,  7 

XIV,  14 

XXX,     30 

C,          100. 

WRITTEN  EXERCISES. 

Express  the  following  numbers  in  figures: 


(1) 

(2) 

(3) 

XIV 

CCL 

MDCL 

XXIV 

DCXC 

MDLX 

XXXIX 

ccxc 

MDLIX 

XCVI 

DCCL 

MDCCC 

CXI 

DCLIX 

MDCCCLX 

CIX 

MCCL 

MDCCCLXIX 

22  INTERMEDIATE  ARITHMETIC. 

Express  the  following  numbers  by  letters : 


(4) 

(5) 

(6) 

(7) 

45 

156 

210 

1500 

76 

184 

550 

1650 

90 

345 

700 

1850 

93 

433 

750 

1868 

99 

555 

880 

1940 

Express  the  following  numbers  by  letters: 

(8)  (9)                                (10)                               (11) 

204  1200                      1685                      2000 

409  1350                      1944                      2050 

540  1408                      1865                      2550 

675  1590                     1909                     3010 


Questions  for  Eeview. 

What  is  arithmetic  ?  What  is  a  number?  What  is  a  unit? 
What  is  an  integer? 

In  how  many  ways  may  numbers  be  expressed  ?  How  are 
numbers  expressed  in  the  Arabic  method?  In  the  Eoman 
method  ?    What  is  notation  ?    What  is  numeration  ? 

How  many  figures  are  used  to  express  numbers?  Which 
are  called  significant  figures?  Which  figure  has  no  numerical 
value  ? 

What  is  meant  by  orders  of  units?  How  are  the  orders 
numbered  ?  How  many  units  of  any  order  make  one  unit 
of  the  next  higher  order? 

What  is  meant  by  the  simple  value  of  a  figure  ?  On  what 
does  the  local  value  of  a  figure  depend?  What  is  the  law  of 
increase  from  right  to  left? 

How  many  orders  make  a  period?  What  are  the  names 
of  these  orders?  Give  the  names  of  the  first  six  periods. 
Give  the  rule  for  notation.     Give   the   rule   for  numeration. 

How  are  numbers  expressed  in  the  Koman  notation? 
Name  the  letters  used,  and  give  the  value  of  each.  How 
are  numbers  expressed  by  these  letters? 


SECTIOlSr    II. 

AD'DITIOJy. 


LESSON   I. 
Add/Jire  JVumhers,   /,  2,  and  3, 

1.  Four  and  2  are  how  many?  8  and  2?  6  and 
2?     7  and  2?     3  and  2?     9  and  2? 

2.  Two  and  3  are  how  many?  5  and  3?  6  and 
3?     7  and  3?     9  and  3?     11  and  3? 

3.  How  many  are  8  and  3?     18  and  3?     38  and  3? 

47  and  3?    67  and  3?     87  and  3? 

4.  How  many  are  9  and  2?     39  and  2?     59  and  2? 

48  and  3?    38  and  3?'    88  and  3? 

5.  Frank  has  5  marbles  in  one  hand  and  2  mar- 
bles in  the  other:  how  many  has  he  in  both  hands? 

Solution. — 5  marbles  and  2  marbles  are  7  marbles :  Frank  has 
7  marbles  in  both  hands. 

6.  A  drover  bought  9  sheep  of  one  farmer  arid  2 
sheep  of  another:    how  many  sheep  did  he  buy? 

7.  Jane  spelled  17  words  correctly  and  mis-spelled 
3:  how  many  words  did  she  try  to  spell? 

8.  A  grocer  sold  8  pounds  of  sugar  to  one  cus- 
tomer, 3  pounds  to  another,  and  2  pounds  to  another : 
how  many  pounds  of  sugar  did  he  sell? 

9.  A  man  walked  4  miles  the  first  hour,  3  miles 
the  second,  and  2  miles  the  third :  how  many  miles 
did  he  walk  in  the  three  hours? 

10.  Eegin  with  1  and  count  to  45  by  adding  2  suc- 
cessively, thus:  1,  3,  5,  7,  9,  11,  13,  etc. 

11.  Begin  with  2  and  count  to  50  by  adding  2  suc- 
cessively;   by  adding  3  successively. 

(23) 


24  INTERMEDIATE  ARITHMETIC. 

WRITTEN  EXERCISES. 
Add  the  following  numbers: 

(1)  (2)  (3)  4)  (5) 

2  10  112  2112  12102 

2  21  211  1201  21210 

1  12  122  1122  10222 

2  20  111  2021  11121 
1  22  222  1212  21212 
L  11  1?L  2221  12111 

(6)  Write  the  numbers  so  that  the  units  shall 

PROCESS.  ^*^^"^  ^^®   ^^'^^  column  ;    the  tens,  the  second 

column ;  and  the  hundreds,  the  third  column. 
121  Begin  with  the  units'  column,  and  add,  nam- 

2'^'^  ing   results  only,  thus:    3,   5,  8,    11,    12,    14, 

12^  17,  20,  21,-21  units  equal  2  tens  and  1  unit. 

^•^-'  AYrite  the  1  unit  under  the  units'  column,  and 

2^1  add  the  2  tens  with  the  tens'  column,  thus:  5, 

^23  8,  10,  12,  15,  18,  20,  23,  25,-25  tens  equal  2 

1"^^  hundreds  and  5  tens.     Write  the  5  tens  under 

the  tens'  column,  and  add  the  2  hundreds 
'^^  with  the  hundreds'  column,  thus:  5,  7,  8,  1 1 


2051,  Sum.        13,  16,  17,  19,  20,-20  hundreds  equal  2  thou- 
sands and  0  hundreds.     Write  the  0  hundreds 
under  the  hundreds'  column,  and  write   the  2  thousands  in 
thousands'  place.     The  sum  is  2051.     To  test  the  accuracy  of 
the  work,  add  the  columns  downward. 


(7) 

(8) 

(9) 

(10) 

13 

232 

1323 

3232 

22 

123 

2112 

2323 

20 

212 

2131 

23213 

81 

131 

3213 

13221 

12 

120 

1301 

32233 

21 

102 

2222 

232111 

23 

223 

nil 

323212 

32 

121 

1323 

232021 

ADDITION.  25 

11.  Add  213,  322,  203,  312,  222,  321,  231,  123,  303, 
232,  311,  132. 

12.  What  is  the  sum  of  2132,  3113,  2323,  1313,  2132, 
and  3320  ? 

13.  2021  +  12333  +  22031  +  332231  +  231323  =  how 
many? 

14.  3231  +  2302  +  2330  +  12332  =  how  many  ? 

15.  A  grocer  sold  12  pounds  of  sugar  to  one  cus- 
tomer, 21  pounds  to  another,  32  pounds  to  another, 
and  30  pounds  to  another:  how  many  pounds  did  he 
sell? 

16.  July  has  31  days;  August,  31;  September,  30; 
October,  31 ;"  November,  30 ;  and  December,  31 :  how 
many  days  in  the  last  six  months  of  the  year? 

17.  A  farm  contains  120  acres,  another  212  acres, 
another  133  acres,  and  another  322  acres:  how  many 
acres  do  the  four  farms  contain? 

18.  A  man  bought  four  loads  of  hay,  the  first  weigh- 
ing 2130  pounds,  the  second  2312  pounds,  the  third 
2232  pounds,  and  the  fourth  2322  pounds :  how  many 
pounds  of  hay  in  the  four  loads? 


LESSON    II. 

JVeff  Additive  J\^ umbers,  4  aiid  5. 
MENTAL   EXERCISES. 

1.  Three  and  4  are  how  many?     5  and  4?     6  and  4? 
8  and  4?     7  and  4?     9  and  4? 

2.  Two  and  5  are  how  many?     4  and  5?     6  and  5? 
8  and  5?    7  and  5?     9  and  5? 

3.  How  many  are  18  and  4?     28  and  4?     48  and  4? 
IG  and  4?     36  and  4?     56  and  4? 


26 


INTERMEDIATE  ARITHMETIC. 


4.  How  many  are  17  and  5?  27  and  5?  47  and  5? 
29  and  57     49  and  5?     69  and  5? 

5.  There  are  17  birds  on  one  tree  and  4  on  another: 
how  many  birds  on  both  trees? 

6.  A  man  gave  26  dollars  for  a  coat  and  5  dollars 
for  a  hat :    how  many  dollars  did  he  give  for  both  ? 

7.  A  drover  bought  19  cows  of  one  man  and  4  of 
another:    how  many  cows  did  he  buy? 

8.  James  picked  27  peaches  from  one  limb  and  5 
peaches  from  another :  how  many  peaches  did  he 
pick  from  both  limbs? 

9.  Mary  has  written  16  lines:  if  she  write  5  lines 
more,  how  many  lines  will  she  then  have  written? 

10.  George  gave  15  cents  for  a  slate  and  5  cents 
for  a  pencil:    how  many  cents  did  he  give  for  both? 

11.  Begin  with  2  and  count  to  50,  or  more,  by 
adding  4  successively;    by  adding  5  successively. 

12.  Begin  with  3  and  count  to  48  by  adding  5 
successively. 

AT^RITTEN  EXERCISES. 


(1) 

(2) 

(3) 

(4) 

(5) 

15 

251 

15215 

23512 

52134 

25 

153 

14343 

30425 

34445 

35 

354 

45046 

41341 

53054 

45 

452 

50350 

23301 

44052 

55 

355 

33432 

41545 

25253 

45 

254 

43543 

43453 

34545 

35 

555 

23343 

25445 

41534 

25 

444 

45452 

41505 

22335 

6.  What  is  the  sum  of  four  hundred  and  four;  four 
thousand  and  forty;  forty  thousand  four  hundred; 
and  four  million  four  hundred  thousand? 


ADDITION.  27 

7.  A  grain  dealer  bought  2350  bushels  of  wheat 
on  Monday,  4215  bushels  on  Tuesday,  3245  bushels 
on  Wednesday,  1500  bushels  on  Thursday,  2424 
bushels  on  Friday,  and  1350  bushels  on  Saturday : 
how  many  bushels  did  he  buy? 

8.  In  a  city  containing  ^ve  wards,  there  are  345 
voters  in  the  first  ward,  443  in  the  second,  213  in 
the  third,  523  in  the  fourth,  and  425  in  the  fifth: 
how  many  voters  in  the  city?  * 

9.  A  father  gave  to  his  eldest  son  225  acres  of 
land,  to  the  second  155  acres,  to  the  third  145  acres, 
and  to  the  youngest  124  acres:  how  many  acres  did 
he  give  to  all? 

10.  The  first  three  cars  of  a  freight  train  contain 
35240  pounds  each ;  the  next  four  cars,  25345  pounds 
each ;  the  next  two  cars,  31540  pounds  each ;  and 
the  last  car,  25432  pounds:  how  many  pounds  of 
freight  in  the  ten  cars? 


LESSON    III. 

JV^^  A  Mi/lye  J^iimher,   6. 
MENTAL  EXERCISES. 

1.  Two  and  6  are  how  many?     4   and    6?     3   and 
6?     5  and  6?     7  and  6?     9  and  6?     8  and  6? 

2.  How  many  are   17  and   6?     28  and  6?     48  and 
6?     68  and  6?     58  and   6?     78  and  6? 

3.  How  many  are  19  and  6?     29  and   6?     59  and 
6?     39  and  G?     69  and  6?     49  and  6? 

4.  Begin    with    3    and    count   to    63    by    adding    6 
successively. 

5.  Mary's  father  gave  her  5  peaches  and  her  mollier 
gave  her  6:    how  many  peaches  did  both  give  her? 


28 


INTERMEDIATE   ARITHMETIC. 


6.  John  solved  18  problems  before  school  and  6 
problems  in  school:  how  many  problems  did  he  solve? 

7.  A  farmer  bought  a  cow  for  27  dollars  and  a  calf 
for  6  dollars:  how  many  dollars  did  he  pay  for  both? 

8.  The  head  of  a  fish  is  5  inches  long,  its  body 
16  inches,  and  its  tail  6  inches:  how  long  is  the 
fish? 

9.  In  a  certain  orchard  there  are  29  apple  trees, 
5  pear  trees,  and  6  peach  trees :  how  many  trees  in 
the  orchard? 

10.  William  gave  a  blind  boy  19  cents,  John  gave 
him  15  cents,  and  Charles  6  cents :  how  many  cents 
did  they  all  give  him? 

WRITTEN  EXERCISES. 


(1) 

(2) 

(3) 

(4) 

(5) 

3640 

24137 

43260 

35260 

305129 

2566 

16126 

32345 

16165 

224603 

1654 

20050 

16606 

32542 

350164 

2366 

16654 

46060 

36344 

255234 

3456 

33456 

50050 

24030 

145344 

5634 

44162 

16566 

33246 

242456 

4565 

23206 

24656 

21438 

145346 

5656 

36562 

32562 

44546 

200500 

6.  Add  thirty-six  thousand  three  hundred  and 
twenty-five;  fourteen  thousand  and  forty-six;  twenty- 
three  thousand  four  hundred  and  five ;  fifteen  thou- 
sand and  sixteen;  and  three  hundred  and  six  thou- 
sand three  hundred  and  four. 

7.  What  is  the  sum  of  three  million  one  thousand 
and  fifty-six;  six  hundred  thousand  six  hundred  and 
twenty -five;  four  million  forty-two  thousand  and  four; 
forty -five  million  six  hundred  and  fifty  thousand? 


ADDITION.  29 


LESSON    IV. 

JV*eH^  Additive  JViember,  7. 
MENTAL  EXERCISES. 

1.  Two  and  7  are  how  many?  5  and  7?  3  and 
7?     6  and  7?     8  and   7  ?     7  and   7?     9  and  7? 

2.  How  many  are  18  and  7?  48  and  7?  68  and 
7?     88  and  7?     28  and  7? 

3.  Fifteen  and  7  arc  how  many?  35  and  7?  65 
and  7?    45  and  7?     75  and  7? 

4.  Begin  with  4  and  count  to  53  by  adding  7 
successively. 

5.  Charles  had  6  marbles  and  his  father  gave  him 
7 :   how  many  marbles  had  he  then  ? 

6.  A  garden  contains  19  pear  trees  and  7  peach 
trees:    how  many  trees  in  the  garden? 

7.  A  man  bought  a  set  of  harness  for  37  dollars 
and  a  saddle  for  7  dollars:  how  much  did  he  \)ixj 
for  both? 

8.  Mr.  Jones  gave  8  plums  to  John,  6  to  Henry, 
and  7  to  George :  how  many  plums  did  he  give  to 
the  three  boys? 

9.  Frank  gave  10  cents  for  a  lead -pencil,  5  cents 
for  a  piece  of  rubber,  and  7  cents  for  paper:  how 
much  did  the  three  articles  cost? 

10.  A  gentleman  gave  36  dollars  for  a  suit  of  clothes, 
7  dollars  for  a  pair  of  boots,  and  5  dollars  for  a  hat : 
how  much  did  he  pay  for  all? 

11.  Count  by  7's  from  2  to  72;  from  5  to  89; 
from  .4  to  95 ;   from  6  to  97. 

12.  Count  by  6's  from  3  to  75;  from  4  to  88; 
from  5  to  95;    from  7  to  97. 


30 


INTERMEDIATE  ARITHMETIC. 


AVKITTEN   EXERCISES. 


(1) 

(2) 

(3) 

(4) 

10640 

24045 

32620 

7121365 

14075 

14036 

75437 

2171634 

26507 

25507 

50743 

1237773 

16021 

46364 

64017 

7143656 

34412 

54563 

32516 

2674467 

53452 

16057 

18416 

6734765 

26123 

72027 

13673 

6574636 

16021 

47735 

31654 

7147347 

5.  What  is  the  sum  of  sixteen  million  four  thou- 
sand and  sixty-five ;  three  hundred  thousand  two  hun- 
dred and  fifty-six;  seven  thousand  and  forty;  and  five 
million  five  thousand  and  seven? 

6.  What  is  the  sum  of  forty-five  million  seven  thou- 
sand and  seventy;  six  million  sixty-five  thousand  two 
hundred  and  six ;  and  seventy-five  thousand  and  forty- 
four? 

7.  January  has  31  days;  February  (except  in  leap 
year),  28;  March,  31;  April,  30;  May,  31;  and  June, 
30:  how  many  days  in  the  first  six  months  of  the 
year  ? 

8.  A  gentleman  owns  five  farms,  containing,  re- 
spectively, 285  acres,  345  acres,  146  acres,  438  acres, 
and  248  acres:  how  many  acres  of  land  does  he 
own  ? 

9.  A  newsboy  sold  327  papers  in  April,  465  in  May, 
318  in  June,  and  278  in  July:  how  many  papers  did 
he  sell  in  the  four  months? 

10.  The  first  ward  of  a  city  contains  1675  youth  of 
school  age  ;  the  second,  2357  youth  ;  the  third,  2347 ; 
the  fourth,  3270;  and  the  fifth,  2677:  how  many  youth 
of  school  age  in  the  city? 


ADDITION.  31 

LESSON    V. 
JVeH'  AddU/'ye  JVumber,    8. 

MENTAL    EXERCISES. 

1.  Two  and  8  are  bow  many?  5  and  8?  3  and  8? 
6  and  8?     4  and  8?     9  and  8? 

2.  How  many  are  16  plus  8?  36  plus  8?  56  plus  8? 
25  plus  8?    45  plus  8?     65  plus  8? 

3.13  +  8?     33  +  8?     53+8?     29  +  8?     49  +  8? 

4.  Count  by  8^8  from  3  to  54;   from  5  to  93. 

5.  Jane  solved  17  problems  in  tbe  morning  and  8 
in  the  evening:  how  many  problems  did  she  solve? 

6.  A  farmer  raised  16  loads  of  wheat  in  one  field 
and  8  loads  in  another:  how  much  wheat  did  he  raise? 

7.  Kate  spelled  38  words  correctly  and  mis-spelled 
8:  how  many  words  did  she  try  to  spell? 

8.  Charles  gave  25  cents  for  a  speller  and  8  cents 
for  a  pencil:  how  much  did  he  give  for  both? 

9.  A  lady  paid  27  dollars  for  a  shawl,  8  dollars  for 
a  bonnet,  and  3  dollars  for  a  pair  of  shoes:  how  much 
did  she  pay  for  all? 

10.  A  merchant  sold  18  yards  of  muslin  to  one  cus- 
tomer, 7  yards  to  another,  and  8  yards  to  another: 
how  many  yards  did  he  sell? 

WRITTEN    EXERCISES. 


(1) 

308 

(2) 

2617 

(3) 

19864 

(4) 

42764 

(5) 
5868 

280 

4565 

34687 

38768 

4384 

667 

6387 

46768 

34187 

5065 

444 

7836 

65837 

63506 

6008 

555 

bm^ 

80040 

24483 

4873 

371 

4084 

18608 

43832 

8345 

736 
644 

8168 
7846 

36084 

45687 

41608 

37860 

6654 
5636 

32  INTERMEDIATE   ARITHMETIC. 

6.  Add  thirty  thousand  six  hundred  and  fifty; 
fifty  thousand  and  eighty-five;  four  hundred  thou- 
sand six  hundred  and  seven ;  and  three  hundred 
and   forty    thousand    and   seventy. 

7.  Add  eight  million  eight  thousand  and  eight; 
eighteen  million  eighteen  thousand  and  eighteen ; 
and  eight  hundred  million  eight  hundred  thousand 
eight   hundred. 

8.  The  distance  by  railroad  from  Philadelphia  to 
Harrisburg  is  106  miles;  from  Harrisburg  to  Pitts- 
burgh, 249  miles;  from  Pittsburgh  to  Crestline,  188 
miles;  from  Crestline  to  Fort  Wayne,  132  miles; 
from  Fort  Wayne  to  Chicago,  148  miles:  how  far 
is  it  from  Philadelphia  to  Chicago? 

9.  One  of  the  wards  of  a  certain  city  contains 
1384  houses;  another,  2868  houses;  another,  857 
houses;  and  another,  1486  houses:  how  many  houses 
in  the  city? 

10.  A  steam-ship  sailed  217  miles  the  first  day;  265 
miles  the  second ;  227  miles  the  third ;  187  miles  the 
fourth ;  and  168  miles  the  fifth ;  how  many  miles  did 
it  sail  in  the  five  days? 


LESSON    VI. 
JVen^  A.dditlre  JSTianbery  9, 

MENTAL   EXERCISES. 

1.  Three  and  9  are  how  many?     7  and  9?     9  and 
7?     8   and   9?     9   and    8?     5    and   9? 

2.  How   many  are    14  +  9?      24  +  9?      44  +  9? 
16  +  9?     36  +  9?     56  +  9? 

3.  How    many  are    17  +  9?      37  +  9?      57  +  9? 
23  +  9?     43  +  9?     63  +  9? 


ADDITION.  33 

4.  Begin  with  3  and  count  to  57  by  adding  \). 

5.  A  farmer  sold  6  hogs  to  his  neighbor  and  9 
to  a  drover:    how  many  hogs  did  he  sell? 

6.  Andrew  sold  8  bunches  of  grapes  and  had  9 
bunches  left:    how  many  bunches  had  he  at  first? 

7.  There  are  17  cows  in  one  field  and  9  cows  in 
another:    how  many  cows  in  both  fields? 

8.  A  pole  is  7  feet  in  the  water  and  9  feet  in  the 
air :    how  long  is  the  pole  ? 

9.  A  man  paid  23  dollars  for  a  coat,  9  dollars  for 
a  pair  of  pants,  and  8  dollars  for  a  vest:  how  much 
did  he  pay  for  the  suit? 

10.  A  boy  paid  45  cents  for  a  ball,  8  cents  for 
marbles,  and  7  cents  for  an  orange :  how  much  did 
he  pay  for  all? 

WRITTEN  EXERCISES. 


(1) 

57384 

(2) 

4369 

(3) 

45566 

48 

(5) 

4868 

5834 

13846 

806 

76 

3769 

691 

3482 

9376 

287 

1804 

2637 

691 

2038 

80 

786 

13484 

5873 

4056 

409 

5863 

596 

578 

8705 

96 

4836 

43486 

509 

6508 

378 

3988 

G.  What  is  the  sum  of  nine  billion  nine  million 
and  nine;  nine  hundred  million  nine  hundred  thou- 
sand nine  hundred  ;  and  ninety  million  nine  hundred 
thousand  and  ninety? 

7.  The  State  of  Maine  contains  31766  square  miles ; 
New  Hampshire,  9280  square  miles;  Vermont,  10212; 
Massachusetts,  7800;  Connecticut,  4674;  and  Rhode 
Island,  1306.  How  many  square  miles  in  all  of  the 
New  England  States? 
I.  A.— 3. 


34  INTERMEDIATE   ARITHMETIC. 

8.  The  distance  by  riiilroud  from  Boston  to  Spring- 
field is  98  miles ;  from  Springfield  to  Albany  103 
miles;  from  Albany  to  Buffalo,  298  miles;  from  Buf- 
falo to  Cleveland,  183  miles ;  from  Cleveland  to  Chi- 
cago, 355  miles.     How  far  from  Boston  to  Chicago? 


LESSON    VII. 

1.  An  orchard  contains  25  apple  trees  and  8  peach 
trees :    how  many  trees  in  the  orchard  ? 

2.  A  gardener  sold  17  quarts  of  strawberries  in 
market  and  9  quarts  to  a  grocer :  how  many  quarts 
did  he  sell? 

3.  A  lady  gave  15  cents  for  tiiread,  8  cents  for 
needles,  and  7  cents  for  pins:  how  many  cents  did 
she  spend? 

4.  James  gave  8  cherries  to  George,  7  to  William, 
6  to  Thomas,- 9  to  Harry,  and  kept  5:  how  many 
cherries  had  he  at  first? 

5.  A  gentleman  gave  95  dollars  for  a  horse,  15 
dollars  for  a  saddle,  and  5  dollars  for  a  bridle :  how 
much  did  he  pay  for  all? 

6.  Begin  with  2  and  add  to  72  by  7'8,  thus:  9, 
16,  23,  30,  37,  etc. 

7.  Begin  with  5  and.  add  to  61  by  8'8. 

8.  Begin  with  3  and  add  to  69  by  6'8. 

9.  Begin  with  4  and  add  to  67  by  9'8. 

WKITTEN   EXERCISES. 

1.  32545  +  8607  +  11709  -f  50063  r^  how  many? 


ADDITION.  35 

2.  A  man  paid  $3575  for  a  lot,  $5450  for  a  house, 
$875  for  a  stable,  and  $675  for  other  improvements : 
what  did  the  property  cost  him  ? 

Note. — This  character  ($)  denotes  dollars,  and  is  called  the  dol- 
lar sign:   $35  is  read  35  dollars;  $1  is  read  1  dollar. 

3.  The  first  book  of  a  series  contains  328  pages; 
the  second,  392  pages;  the  third,  400  pages;  and  the 
fourth,  432  pages:  how  many  pages  in  the  series? 

4.  Ohio  contains  39964  square  miles;  Michigan, 
56243  square  miles;  Indiana,  33809  square  miles;  and 
Illinois,  55409  square  miles:  what  is  the  area  of  these 
four  States? 

5.  The  distance  by  railroad  from  Pittsburgh  to  Co- 
lumbus is  193  miles;  from  Columbus  to  Cincinnati, 
120  miles;  from  Cincinnati  to  St.  Louis,  340  miles: 
how  far  is  it  from  Pittsburgh  to  St.  Louis? 

6.  A  father  divided  his  estate  between  two  sons 
and  three  daughters,  giving  to  each  son  $3250,  and 
to  each  daughter  $2750 :  what  was  the  value  of  the 
estate  ? 

7.  A  farmer  raised  in  one  year  380  bushels  of 
wheat,  245  bushels  of  oats,  87  bushels  of  rye,  and  as 
many  bushels  of  corn  as  of  wheat,  oats,  and  rye 
together:  how  many  bushels  of  grain  did  he  raise? 


DEFINITIONS,  PEINOIPLES,  AND  EULE. 

Art.  26.  Addition  is  the  process  of  finding  the  sum 
of  two  or  more  numbers. 

The  number  obtained  by  adding  two  or  more  num- 
bers is  called  the  Sum  or  Amount. 

The  Slim  contains  as  many  units  as  all  the  num- 
bers added,  taken  together. 


36  INTKIIAIKDIATE   ARITHMETIC. 

Numbers  are  either  Concrete  or  Abstract. 

A  Concrete  Number  m  applied  to  a  particular 
thing  or  quantity;  as,  4  pears,  7  hours,  30  stei)s. 

An  Abstract  Number  is  not  applied  to  any  par- 
ticular thing  or  quantity ;    as,  4,  7,  30. 

Fourteen  balls  and  13  balls  are  numbers  of  the  same  kind; 
and  6  tens  and  3  tens  are  numbers  of  the  same  order.  Num- 
bers of  the  same  kind  or  order  are  ealled  Like  Numbers. 
Only  like  numbers  can  be  added. 

Art.  27.  The  Sign  of  Addition  is  -f  .  It  is  called 
plus^  meaning  more.  When  placed  between  two  num- 
bers, it  shows  that  they  are  to  be  added.  Thus,  8  -)- 
5  is  read  8  plus  5,  and  it  shows  that  5  is  to  be  added 
to  8. 

The  Sign  of  Equality  is  n= .  It  is  read  equals  or  is 
equal  to.     Thus,  7  -f  8  =  1»  is  read  7  ^)^w.s  8  equals  15. 

Art.  28.  EuLE  for  Addition. — 1.  Write  the  numbers 
to  be  added  so  that  figures  denoting  units  of  the  same 
order  shall  be  in  the  same  column,  and  draw  a  line 
underneath. 

2.  Beginning  with  units,  add  each  column,  and  write 
the  sum,  when  less  than  ten,  underneath. 

3.  When  the  sum  of  any  column  exceeds  nine,  write 
the  right-hand  figure  under  the  column  added,  and  add 
the  number  denoted  by  the  left-hand  figure  or  figures 
with  the  next  column. 

4.  Write  the  entire  surn  of  the  left-hand  column. 
Proof. — Add  the  columns  downward. 


8T]CTTOX   III. 


s  msTHA  cTiojr. 


LESSON    I. 

SubtraJi 671(2  I^igiires^   /_,  2,  S, 

1.  How  many  is  4  less  3?  6  less  3?  8  less  3? 
10  less  3?     12  less  3?     11  less  3? 

2.  How  many  is  11  less  2?  21  less  2?  41  less  2? 
19  less  2?     29  less  2?     49  less  2? 

3.  Three  from  12  leaves  how  many?  3  from  22? 
3  from  42?  3  from  52?  3  from  32?  3  from  20? 
3  from  40?     3  from  50? 

4.  Begin  with  50  and  count  back  to  0  by  sub- 
tracting 2  successively,  thus:    50,  48,  4G,  44,  42,  etc. 

5.  Begin  with  40  and  count  back  to  1  by  sub- 
tracting 3  successively. 

G.  Charles  bought  12  sticks  of  candy  and  ate  3 
of  them :    how  many  sticks  were  left? 

Solution.— 12  sticks  less  3  sticks  are  9  sticks:  Charles  had  9 
sticks  left. 

7.  Henry  wrote  21  words,  but  misspelled  2  of  them : 
liow  many  words  did  he  spell  correctly? 

8.  Charles's  lesson  consists  of  15  examples,  and  he 
has  solved  all  but  3  of  them  :  how  many  has  he  solved? 

9.  James  is  11  years  old,  and  his  brother  Henry  is 

3  years  younger:  how  old  is  Henry? 

(37) 


^^8  INTERMEDIATE    ARITHMETIC. 

WRITTEN   EXERCISES. 

1.  From  345  take  123. 

PROCESS.  Write  123  under  345,  placing  units  un- 

Minuendj  345  ^^^  units,  tens  under  tens,  and  hundreds 
Subtrahend  123  under  hundreds.  Subtract  3  units  from 
Difference  222  ^  units,  and  write  2  units,  the  difference, 
below;  subtract  2  tens  from  4  tens,  and 
write  2  tens,  the  difference,  below;  sub- 
tract 1  hundred  from  3  hundreds,  and  write  2  hundreds,  the 
difference,  below.     The  difference,  or  remainder,  is  222. 


(2) 

(13) 

(4) 

(5) 

(6) 

(7) 

(8) 

57 

46 

88 

75 

685 

409 

967 

43 

24 

65 

53 

343 

307 

645 

(9) 

(10) 

(11) 

(12) 

(13) 

(14) 

(15) 

246 

487 

507 

718 

563 

485 

560 

132 

231 

302 

312 

330 

212 

320 

16.  From    four   thousand    and    sixty-five    take   two 
thousand   and   thirty-one. 

17.  A  grocer  bought  585  pounds  of  sugar  and  sold 
231  pounds:    how  many  pounds  had  he  left? 

18.  In  a  graded  school,  there  are  345  boys  and  321 
girls:  how  many  more  boys  than  girls  in  the  school? 

LESSON    II. 

JVen'  Subtrahend  li*tgte7^es,  Z-  and  5. 

MENTAL  EXERCISES. 

1.  How   many   is   7   less  4?     6   less  4?     9   less  4? 
8   less  4?     10   less  4?     11    less   4? 

2.  How   many   is   7   less   5?     9   less   5?     8   less  5? 
10   less   5?     12   less   5?     15   less   5? 


SUBTRACTION.  39 

3.  How  many  is  13  less  4?  23  less  4?  43  less 
4?     63   less   4?     83   less  4?     53   less  4?     93  less  4? 

4.  How  many  is  14  less  5  ?  44  less  5  ?  34  less 
5?     54   less   5?     64   less   5?     74   less   5  ?     94  less  5? 

5.  Begin  with  60  and  count  back  to  0  by  sub- 
tracting 4  successively. 

6.  Begin  with  53  and  count  back  to  1  by  sub- 
tracting 4  successively. 

7.  A  man  gave  $12  for  a  saddle  and  $4  for  a  bridle: 
how  much  did  the  saddle  cost  more  than  the  bridle? 

8.  Charles  earned  21  cents  by  selling  papers,  and 
gave  4  cents  for  a  comb :  how  many  cents  had  ho 
Mt? 

9.  Kate  is  15  years  old  and  her  sister  is  4  years 
younger:    what  is  her  sister's  age? 

10.  There  are  21  passengers  in  a  car:  if  5  of 
them  leave  at  a  station,  how  many  will  reraain? 

11.  There  are  13  men  in  one  coach  and  5  men  in 
another:  how  many  men  in  the  first  coach  more 
than    in   the    second? 

WRITTEN  EXERCISES. 


(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

335 

2036 

308 

1565 

3683 

7863 

214 

1034 

205. 

1433 

2542 

4552 

7.  From  five  thousand  and   seventy-six  take   threo 
thousand  and  fifty. 

8.  A  farm  contains  358  acres  of  land :    if  155  acres 
should  be  sold,  how  many  would  be  left? 

9.  A  man  bought  a  house  for  $4320  and  sold  it  for 
$6450:    how  much  did  he  gain? 

10.  A  man  bought  3487  bushels  of  wheat  and  sold 
1425  bushels:    how  many  bushels  had  he  left?- 


40  INTERMEDIATE  ARITHMETIC. 

11.  A  ship-builder  sold  a  vessel  for  $24350:  if  tlie 
vessel  cost  him  $27585,  how  much  did  he  lose? 

12.  The  number  of  school-houses  in  Ohio,  in  1867, 
was  11353;  in  Pennsylvania,  11453:  how  many  more 
school-houses  in  Pennsylvania  than  in  Ohio  ? 

13.  A  wool-dealer  having  bought  23437  fleeces  of 
wool,  shipped  12322  fleeces  to  Boston :  how  many 
fleeces  had  he  left? 

LESSON    III. 

jy^^'ft^  Subh^ahend  J^lgtij^es,  6  and  7. 

MENTAL  EXERCISES. 

1.  How  many  is  8  less  6?  10  less  0?  12  less  G? 
13   less    6?     15    less    0?     14    less    (>? 

2.  How  many  is  12  less  7?  13  less  7?  15  less  7? 
16   less   7?     18   less   7?     22   less   7? 

3.  How  many  is  14  less  6?  24  less  6?  44  less  6? 
64   less   6?     34   less   6?     74   less   6? 

4.  How  many  is  14  less  7?  24  less  7?  44  less  7? 
16   less   7?     36   less   7?     46   less   7? 

5.  Begin  with  56  and  count  back  to  0  by  7's. 

6.  Begin  with  60  and  count  back  to  0  by  6's. 

7.  Ella  was  absent  from  school  7  days  in  a  term 
of  75  days:  how  many  days  was  she  present? 

8.  John  earned  25  cents  by  selling  oranges,  and  gave 
6  cents  for  a  pencil:  how  many  cents  had  he  left? 

9.  A  boy  was  carrying  home  21  eggs  ;  he  fell,  and 
broke  7  of  them:  how  many  were  left? 

10.  A  man,  having  23  dollars,  gave  6  dollars  for 
a  hat:  how  many  dollars  had  he  left? 

11.  A  teacher  pronounced  25  words  to  an  idle  pupil, 
who  mis-spelled  7  of  them :  how  many  words  did  he 
spell  correctly  ? 


SUBTPv  ACTION.  41 

WKITTEN   EXERCISES. 

1.  From  5334  take  2726. 

PROCESS.  Since  6  units  can  not  be  taken  from  4  units, 

Mm.  5334  ^^^  1^  units  to  the  4  units,  making  14  units; 
Sub.  2726  then  subtract  6  units  from  14  units,  and  write 
Dif  2608  ^  units,  the  difference,  below.  To  balance  the 
10  units  (equal  1  ten)  added  to  the  minuend, 
add  1  ten  to  the  2  tens  of  the  subtrahend; 
then  subtract  3  tens  from  8  tens,  and  write  0  tens,  the  differ- 
ence, below. 

Add  10  hundreds  to  the  8  hundreds  of  the  minuend,  mak-'^ 
ing  13  hundreds;  subtract  7  hundreds  from  13  hundreds,  and 
write  6  hundreds,  the  difference,  below.  To  balance  the  10 
hundreds  (equal  1  thousand)  added  to  the  minuend,  add  1 
thousand  to  the  2  thousands  of  the  subtrahend ;  subtract  3 
thousands  from  5  thousands,  and  write  2  thousands,  the  dif- 
ference, below.     The  difference  is  2608. 

This  process  may  be  shortened,  thus :  6  units  from  4  units 
plus  10  units,  or  14  units,  leave  8  units;  2  tens  and  1  ten  are 
3  tens,  and  3  tens  from  3  tens  leave  0  ten ;  7  hundreds  from 
3  hundreds  plus  10  hundreds,  or  13  hundreds,  leave  6  hun- 
dreds; 1  thousand  and  2  thousands  are  8  thousands,  and  8 
thousands  from  5  thousands  leave  2  thousands.  The  differ- 
ence is  2608. 

NoTp]. — The  teacher  should  show  that  the  adding  of  10  to  a  term 
of  the  minuend  and  1  to  the  next  higher  term  of  the  subtrahend 
increases  both  minuend  and  subtrahend  equally,  and  does  not 
affect  the  difference. 


C^) 

(3) 

(4) 

(5) 

(6) 

44 

63 

272 

1385 

5754 

26 

46 

147 

1276 

3457 

(7) 

(8) 

(9) 

(10) 

(11) 

3416 

3041 

14406 

20670 

30401 

2507 

2637 

7345 

17856 

20576 

42  INTERMEDIATE  APvITHMETIC. 

12.  From  fourteen  thousand  and  forty-four  take 
six   thousand    and   sixteen. 

13.  A  man  whose  income  is  $1850  expends  annually 
$1365:  how  much  does  he  lay  up? 

14.  The  number  of  youth  of  school  age  in  a  certain 
city  is  1234,  and  only  756  pupils  are  enrolled  in  the 
schools :  how  many  youth  do  not  attend  school  ? 

15.  The  number  of  pupils  enrolled  in  the  public 
schools  of  Ohio,  in  1867,  was  704767;  in  Pennsyl- 
vania, 789389:  how  many  more  pupils  were  enrolled 
in  Pennsylvania  than  in  Ohio? 

LESSON    IV. 

JVen^  S/ibirahe7id  J^igiires,  8  and  9, 
MENTAL    EXEKCISES. 

1.  How  many  is  9  less  8?  11  less  8?  13  less  8? 
10  less  8?     14  less  8?     12  less  8? 

2.  How  many  is  16  less  8?  26  less  8?  56  less  8? 
17  less  8?     27  less  8?     67  less  8? 

3.  How  many  is  11  less  9?  13  less  9?  15  less  9? 
12  less  9?     16  less  9?     17  less  9? 

4.  How  many  is  16  less  9?  26  less  9?  36  less  9? 
15  less  9?     25   less  9?     45  less  9? 

5.  Begin  with  50  and  count  back  to  2  by  subtract- 
ing 8  successively. 

6.  Begin  with  57  and  count  back  to  3  by  subtract- 
ing 9  successively. 

7.  A  school  has  enrolled  65  pupils,  and  8  are  ab- 
sent: how  m^ny  are  present? 

8.  Mr.  Smith  is  44  years  of  age  and  his  ^"oungest  son 
is  8  years  of  age:  what  is  the  ditference  in  their  ages? 

9.  A  school  contains  9  more  girls  than  boys :  if 
there   are   56   girls,  what   is   the    number   of    boys? 


SUBTRACTION.  43 

WBITTEN    EXERCISES. 

1.  From  800000  take  238. 

2.  From  forty  million  take  eighty  thousand. 

3.  A  nursery  contains  705  peach  trees  and  428 
plum  trees:  how  many  more  peach  trees  than  plum 
trees  in  it? 

4.  The  Pilgrims  landed  at  Plymouth  in  1620,  and 
our  National  Independence  was  declared  in  1776 : 
how  many  years  between  the  two  events? 

5.  The  first  steam-boat  was  made  in  1807,  and  the 
Atlantic  Cable  was  laid  in  1866 :  how  many  j^ears 
between  the  two  events? 

6.  Mont  Blanc  in  Europe  is  15668  feet  high,  and 
Mount  Sorata  in  South  America  is  21286  feet  high : 
what  is  the  difference  in  the  height  of  these  two 
mountains? 

7.  A  man  who  owned  3408  sheep,  sold  1897  of 
them:    how  many  sheep  had  he  left? 

8.  Mt.  Etna  is  10874  feet  high,  and  Mt.  Yesuvius 
3948  feet:   how  much  higher  is  Etna  than  Vesuvius? 

9.  America  was  discovered  in  1492,  and  the  Pil- 
grims landed  at  Plymouth  in  1620:  how  many  years 
intervened? 

10.  The  population  of  the  State  of  New  York  in 
1860  was  3881000,  and  that  of  Ohio,  2340000:  how 
many  more  people  in  New  York  than  in  Ohio? 

LESSON    V. 

1.  From  a  cask  containing  45  gallons  of  molasses, 
39  gallons  were  sold :  how  many  gallons  remained 
unsold  ? 

2.  An  orchard  contains  56  apple  trees  and  48  peach 


44  INTERMEDIATE  ARITHMETIC. 

trees:    how  many  more  i\])])\e  trees  than    peach   trees 
in  the  orchard  ? 

3  A  grocer  sold  57  pounds  of  butter  from  a  firkin 
containing  G5  pounds:  how  many  pounds  remained 
in  the  firkin? 

4.  In  a  school,  63  pupils  are  enrolled  and  54  are 
present:    how  many  pupils  are  absent? 

5.  If  a  man  earn  $45  a  month,  and  spend  $36, 
how  much  does  he  lay  up? 

6.  A  man  gave  $75  for  a  watch  and  $22  for  a 
chain :  how  much  did  the  watch  cost  more  than 
the   chain  ? 

7.  Charles  has  17  marbles  and  John  8 :  how  many 
more  marbles  has  Charles  than  John  ? 

8.  A  teacher  asked  his  class  52  questions,  and  8 
were  answered  incorrectly :  how  many  were  answered 
correctly  ? 

9  In  a  term  of  64  days,  Charles  attended  school 
55  days:  how  many  days  was  he  absent? 

10.  Subtract   by  4's   from  62  back   to   2. 

11.  Subtract  by   6's   from  75   back   to   3. 

12.  Subtract  by   9's   from  68  back   to   5. 

13.  Subtract  by   7's   from  59   back   to   3. 

14.  Subtract   by   8's   from  48   back   to   0. 

\/V^BITTEN  EXERCISES. 

1.  From  202380  take  165436. 

2.  4308560  —  1674805  =  how  many? 

3.  Illinois  contains  55409  square  miles,  and  Missouri 
67380:  how  much  more  area  has  Missouri  than  Illinois? 

4.  By  the  census  of  1850  the  entire  population  of 
the  United  States  was  23191876,  and  by  the  census 
of  1860  it  was  31224885:  what  was  the  increase  in 
10  years? 


SUBTRACTION.  45 

5.  In  1862  there  were  10869  miles  of  railroad  in 
Great  Britain,  and  33222  miles  in  the  United  States: 
how  many  more  miles  in  the  United  States  than  in 
Great  Britain  ? 

6.  An  army  of  30340  men  lost  7568  in  battle:  how 
many  men  did  it  then  contain? 

7.  In  1862  Ohio  produced  35442858  pounds  of  butter 
and  20637235  pounds  of  cheese:  how  much  more  butter 
than  cheese  was  produced? 

8.  A  merchant  having  S11315  in  bank,  drew  out 
$978:    how  much  remained  in  the  bank? 

DEFINITIONS,  PEINCIPLES,  AND  EULE. 

Art.  29.  Subtraction  is  the  process  of  finding  the 
difference  between  two  numbers. 

The  Difference  or  lieniainder  is  the  number 
found  by  subtracting  one   number  from  another. 

The  Minuend  is  the  number  diminished. 

The  SubtrahenU  is  the  number  subtracted. 

Art.  30.  Only  Li  he  JVinnhers  can  be  subtracted. 
Three  pencils  can  not  be  subtracted  from  7  books, 
nor  3  units  from  7  tens. 

Art.  31.  The  Sign  of  Subtraction  is  — .  It  is  read 
minus  or  less.  It  shows  that  the  number  after  it  is 
to  be  subtracted  from  the  number  before  it. 

Art.  32.  EuLE  for  Subtraction.  —  1.  Write  the  sub- 
trahend  under  the  minuend.,  placing  units  under  units, 
tens  under  tens,  hundreds  under  hundreds,  etc. 

2.  Begin  at  the  right,  and  subtract  each  term  of  the 
subtrahend  from  the  term  above  it,  and  icrite  the  differ- 
ence underneath. 


46  INTERMEDIATE  ARITHMETIC. 

3.  When  any  term  of  the  subtrahend  is  greater  than 
the  term  above  it^  add  10  ^o  the  upper  term,  and  then 
subtract,  and  write  the  difference  as  before. 

4.  When  10  has  been  added  to  the  upper  term,  add 
1  to  the  next  higher  term  of  the  subtrahend  before 
subtracting. 

Proof. — Add  the  remainder  and  subtrahend ;  if  their 
sum  is  equal  to  the  minuend,  the  work  is  correct. 

Note. — Instead  of  adding  1  to  the  next  term  of  the  subtrahend, 
1  may  be  subtracted  from  the  next  term  of  the  minuend. 

LESSON    VI. 

'Problems  combining  AddUlon  and  Subhead  ion, 

1.  Eobert  picked  21  peaches,  and  gave  7  to  his 
Bister  and  8  to  bis  brother:  how  many  peaches  had 
he  left? 

2.  A  garden  contains  17  pear  trees,  8  plum  trees, 
and  a  certain  number  of  peach  trees :  if  there  arc 
33  trees  in  the  garden,  what  is  the  number  of  peach 
trees  ? 

3.  A  grocer  bought  35  bushels  of  apples,  and  sold 
17  bushels  to  A,  9  bushels  to  B,  and  the  rest  to  C : 
how  many  bushels  did  he  sell  to  C? 

4.  Jane  is  8  years  old  and  Lucy  13,  and  the  sum 
of  Jane's  and  Lucy's  ages,  less  7  years,  is  the  age 
of  Mary:    how  old  is  Mary? 

5.  A  man  bought  a  firkin  of  butter  for  $17,  a  crock 
of  lard  for  $8,  and  a  barrel  of  flour  for  $9;  but  he 
had  not  money  enough  by  $7  to  pay  for  them :  how 
much  money  had  he? 

6.  A  man  earned  $45,  and  paid  $15  for  house  rent, 
$8  for  flour,  $7  for  shoes,  and  $10  for  groceries :  how 
much  had  he  left? 


SUBTRACTION.  47 

7.  A  man  sees  15  pigeons  on  one  branch  of  a  tree, 
and  9  pigeons  on  another  branch :  if  7  should  fly 
away,  how  many  would  be  left  on  the  tree? 

8.  A  farmer  had  23  chickens,  but  7  of  them  were 
stolen  and  5  were  carried  off  by  a  hawk:  how  many 
chickens  had  he  left? 

9.  A  drover  bought  17  sheep  of  one  farmer,  9  sheep 
of  another,  and  8  of  another,  and  then  sold  7  of  them 
to  a  butcher:  how  many  sheep  had  he  left? 

10.  A  man  gave  a  watch  and  $9  in  money  for  a 
horse  valued  at  $75:   what  did  he  get  for  his  watch? 

WRITTEN  EXERCISES. 

1.  From  a  piece  of  carpeting  containing  150  yards, 
a  merchant  sold  3  carpets,  containing  27,  39,  and  42 
yards,  respectively  :    how  many  yards  were  left  ? 

2.  Ehode  Island  contains  an  'area  of  1306  square 
miles;  Delaware,  2120;  Connecticut,  4674 ;  New  Jer- 
sey, 8320;  Maryland,  9356;  and  New  York,  47000: 
how  many  more  square  miles  has  New  York  than 
the  other  five  States  named? 

3.  A  regiment  entered  the  service  with  1088  men ; 
150  were  killed  in  battle,  65  died  from  disease,  24 
deserted,  and  250  were  discharged :  how  many  re-* 
mained  ? 

4.  A  grain  dealer  bought  1250  bushels  of  wheat  on 
Monday,  2145  bushels  on  Tuesday,  and  3240  bushels 
on  Wednesday,  and  on  Thursday,  fearing  a  decline 
in  price,  he  sold  5450  bushels :  how  much  wheat  had 
he  left? 

5.  A  man  deposited  $175,  $141,  $75,  $304,  and  $250 
in  a  bank,  and  then  drew  out  $480  and  $225:  how 
many  dollars  remained  in  the  bank? 

6.  A   railroad    train    left    Cincinnati    for   St.    Louis 


48  INTERMEDIATE   ARITHMETIC. 

with  336  passengers,  and  during  the  trip  145  passen- 
gers came  aboard,  and  208  passengers  left:  how  many 
were  in  the  cars  when  the  train  reached  St.  Lonis? 

7.  In  1860  the  popuhition  of  Maine  was  628279; 
New  Hampshire,  326073;  Vermont,  315098;  Massa- 
chusetts, 1231066;  Connecticut,  460147;  Khode  Ishind, 
174620;  and  New  York,  3880735:  how  many  more 
inhabitants  in  New  York  than  in  the  six  New  Eng- 
hmd  States? 

8.  A  man  gave  to  his  eldest  son  $2380;  to  the 
second,  $245  less  than  to  the  eldest;  and  to  the 
youngest,  $450  less  than  to  the  second :  how  much 
did  he  give  to  all  ? 

9.  From  the  sum  of  2348  and  1864  subtract  their 
difference. 

10.  From  the  sum  of  506703  and  340067  take  their 
difference. 


SECTION  IV. 
MUZ  TI^JPZ ICA  TIOJV'. 


LESSON    I. 

M'leHf'pllcajfd  I^fgttres,  1 ,  2,  and  S, 

1.  Twice  2  arc  how   many?     4   times   2?     6   times 
2?     5  times  2?     8  times  2?     9  times  2? 

2.  Twice  3  are  how  many?     3   times   3?     5  times 
3?     4  times  3?     7  times  3?     9  times  3? 


MULTirLICATION.  49 

3.  How  many  are  5  times  1  ?     5  times  3  ?     7  times 
1?     7  times  2?     8  times  1?     8  times  3? 

4.  A  boy  has  2   hands:    how  many  hands  have  6 
boys?     8  boys?     10  boys? 

5.  There   are  3  feet  in  a  yard :    how  many  feet  in 
2  yards?     4  yards?     5  yards?     7  yards? 

6.  If  a"  man  earn  3  dollars  a  day,  how  many  dollars 
will  he  earn  in  6  days? 

Solution. — If  a  man  earn  3  dollars  in  one  day,  in  6  days  ho 
will  earn  6  times  3  dollars,  which  is  18  dollars. 

7.  If  a  boy  walk  3  miles  a  day  in  attending  school, 
how  many  miles  will  he  walk  in  10  days? 

WKITTEN   EXERCISES. 

1.  Multiply  232  by  3. 

PROCESS.  Write    the    multiplier    3    under    the 

Multiplicand,  2^2        ^^its'  figure  of   the   multiplicand,   and 

Multiplier,  3         multiply,    thus:    8    times    2    units    are 

P-  <7    /  /^Q/?         6   units;    3   times  3  tens   are    9    tens; 

'  3   times   2   hundreds    are   6    hundreds. 

The   product   is  696. 

(2)  (3)  (4)  (5)  (G) 

3212    10202    23321    202122    303203 
3        3        3         4         3 


7.  Multiply  230321  by  2.     By  3. 

8.  Multiply  320201  by  3.     By  4.     By  2. 

9.  If  a  gold  watch  is  worth  $220,  what  is  the  worth 
of  4  such  gold  watches  ? 

10.  A  drover  bought  3  horses  at  $133  apiece :  what 
did  they  cost? 

11.  There  are  320  rods  in  a  mile:    how  many  rods 
are  there  in  3  miles?     In  4  miles? 

I.  A.— 4. 


50  INTERMEDIATE   ARITHMETIC. 


LESSON    II. 

J^ew  Multiplicand  JFlgures,  J^  and  5, 
MENTAL   EXERCISES. 

1.  Twice  4  are  how  many?     3   times   4?     5  times 
4?     6  times  4?     8  times  4?     7  times  4?     9  times  4? 

2.  Twice  5  are  how  many?     5  times  5?     6   times 
5?     8  times  5?     7  times  5?     9  times  5? 

3.  How  many  are  7  times  4?     7  times  5?     9  times 
4?     9  times  5?     10  times  5? 

4.  If  a  lemon  cost  4  cents,  what  will  6  lemons  cost? 

5.  How  much  will  a  man  earn  in  7  days  at  $4  a 
day?     In  8  days?     In  9  days? 

6.  There   are   5    cents   in   a   half-dime:    how  many 
cents  in  3  half-dimes?     5  half-dimes? 

7.  If  you  write  5  lines  a  day,  how  many  lines  will 
you  write  in  4  days?     In  7  days? 

8.  If  5  boys  can  sit  on   1  bench,  how  many  boys 
can  sit  on  8  benches? 

9.  What  will  10  oranges  cost  at  5  cents  apiece? 

10.  If  there  are  5  school -days  each  week,  how  many 
are  there  in  6  weeks?     In  8  weeks?     In  10  weeks? 

WRITTEN  EXERCISES. 

.     1.  Multiply  434  by  6. 

PROCESS.  Multiply  the  number  denoted  by  each 

Multiplicand,  434         ^g^re  of  the  multiplicand  by  6.     Thus: 
Multiplier  6        ^    times   4    units   are    24   units,    which 

P    rl    f        or 04        Gq^^al   2   tens  and  4  units;    write  the  4 
units  in  units'  place  in  the  product,  and 
reserve  the  2  tens.     Six  times  3  tens  are 
18   tens,  and   18   tens  plus  the  2  tens  reserved  are  20  tens, 


MULTIPLICATION.  51 

which  equals  2  hundreds  and  0  tens;  write  the  0  tens  in  tens* 
place  in  the  product,  and  reserve  the  2  hundreds.  Six  times 
4  liundreds  are  24  hundreds,  and  24  hundreds  plus  the  2 
hundreds  reserved  are  26  hundreds,  which  equals  2  thou- 
sands and  6  hundreds;  write  the  6  hundreds  in  hundreds' 
place  in  the  product,  and  the  2  thousands  in  thousands' 
place.     The  product  is  2604. 


(2) 

(3) 

(4) 

(5) 

(6) 

(7)  ^ 

453 

2524 

4545 

3545 

13545 

25245 

8 

6 

7 

8 

4 

6 

8.  If  there  are  324  pins  on  a  paper,  how  many- 
pins  are  there  on  3  papers?     5  papers? 

9.  If  a  train  of  ears  run  425  miles  a  day,  bow  far 
will  it  run   in   8  days? 

10.  If  135  tons  of  iron  rails  will  make  one  mile  of 
railroad,  how  many  tons  will  make  7  miles? 

11.  What  will  6  horses  cost  at  $152  apiece? 

12.  A  father  divided  his  estate  between  four  sons, 
giviiig  to  each  $3545 :  what  was  the  value  of  the 
estate  ? 

13.  There  are  1440  minutes  in  a  day  :  how  many 
minutes  in  7  days,  or  a  week? 

14.  If  it  take  15520  shingles  to  cover  a  house,  how 
many  shingles  will  it  take  to  cover  8  houses? 


LESSON    III. 

JVe>t^  MuUipUcarid  I^i^ure,   6, 
MENTAL   EXERCISES. 

1.  Twice  6  are  how  many?  4  times  6?  3  times 
6?  5  times  6?  7  times  6?  6  times  6?  8  times  6? 
9  times  6?     10  times  6? 


52  INTERMEDIATE   ARITHMETIC. 

2.  There   are    8  rows  of  trees   in   an  orchard,  and 
G  trees  in  each   row:   how  many  trees  in  the  orchard? 

3.  What  will  7  lead-pencils  cost  at  6  cents  apiece  ? 

4.  What  will   6  oranges  cost  at  8  cents  apiece? 

5.  There  are  6  days  for  labor  in  each  week :    how 
many  days  for  labor  in   6  weeks?     9  weeks? 

6.  John    caught    6    fishes,    and    Harry    7    times    as 
nrany  as   John :    how  many  did   Harry  catch  ? 

7.  If  a  horse  travel  6  miles  an  hour,  how  far  will 
it  travel  in  5   hours?     In   10  hours? 

8.  There  are  6  feet   in   a   fathom:    how  many  feet 
in  7  fathoms?     9  fathoms? 


WRITTEN  EXERCISES. 

1.  Multiply  456  by  43. 

PROCESS.  Write  the  multiplier  under  the  mul- 

Multiplicand,     456  tiplicand,  placing  units  under  units  and 

Multiplier             43  ^^^^  under  tens.     First  multiply  by  the 

P    f  1      (     1  Qr8  ^    units,   as    in    the    preceding    lesson, 

J    J     i  -loo^  which   sfives  1368  for  the   first   partial 

products,  [1824  i     ^      xr     ^        i^-   i    i.  \n      a  ^ 

product.     Next  multiply  by  the  4  tens, 

Product,  19608  o]^)gej,ving  that  units  multiplied  by  tens 
(or  tens  by  units)  produce  tens,  that 
tens  by  tens  produce  hundreds,  and  that  hundreds  by  tens 
produce  thousands,  etc.  This  gives,  for  the  second  partial 
product,  4  te7is,  2  hundreds,  8  thousands,  and  1  ten-thousand, 
which  are  to  be  written  in  their  proper  orders,  since  unlike 
orders  can  not  be  added.  Then  add  the  two  partial  products, 
and  their  sum,  which  is  19608,  is  the  product  required. 

Note. — The  teacher  should  show  that  units  multiplied  hy  tens 
produce  tens;  tens  by  tens,  hundreds,  etc.  This  may  be  done,  in 
the  above  example,  by  changing  the  4  tens  into  40  units.  40 
times  6  units  =  240  units,  or  24  tens;  and  40  times  5  tens  =  200 
tens,  or  20  hundreds,  etc.  The  first  figure  of  each  partial  product 
is  written  under  the  multiplier  which  produces  it. 


MULTIPLICATION.  53 


(2) 

<3) 

(4) 

(5) 

(6) 

(7) 

(8) 

606 

562 

653 

1446 

2306 

4636 

40563 

54 

67 

86 

234 

726 

67 

143 

9.  If  a  ship  sail  216  miles  a  day,  how  far  will  it 
sail  in  38  days? 

10.  What  will  27  carriages  cost  at  $165  apiece? 

11.  If  a  web  of  flannel  contain  46  yards,  how  manj^ 
yards  in  397  webs? 


LESSON   IV. 

JVen^  Multiplicand  I^lgure^    7* 
MENTAL  EXERCISES. 

1.  Three  times  7  are  how  many?  5  times  7?  7 
times  7?     9  times  7?     8  times  7? 

2.  There  are  7  days  in  a  week:  liow  many  days  in 
2  weeks  ?     4  weeks  ? 

3.  How  many  hills  of  potatoes  in  6  rows  if  there 
are  7  hills  in  each  row? 

4.  If  Charles  earn  7  dollars  a  week,  liow  much 
will    ho   earn    in    5  weeks? 

5.  If  a  horse  travel  7  miles  in  an  hour,  how  far 
will   he   travel    in    8   hours? 

6.  If  5  men  can  build  a  wall  in  7  days,  how  long 
will  it  take   1  man  to  build   it? 

7.  If  a  box  of  crackers  will  last  8  men  7  days, 
how  long  will    it   last   1    man  ? 

8.  An  orchard  contains  10  rows  of  trees,  and  there 
are  7  trees  in  each  row :  how  many  trees  in  the 
orchard  ? 


54  INTERMEDIATE  ARITHMETIC. 

WRITTEN     EXERCISES. 

1.  Multiply  2745  by  306. 

PROCESS.  Multiply  successively  by  the   first 

2745        ^^^^  third   figures  of   the   multiplier, 

3Q5        observing    that    units    multiplied    by 

-r»     ,.7     r    TTTT^  hundreds  produce  hundreds,  and  hence 

Partial     (164/0  ^/    .    ^  ^             ^ '  ,                , 

7    ^    ^  o  r»  o  r  writmo;  the  first  figure  or  the  second 

2)roducfs,  (8235  .f        j     ^    -      u      a     a  ^        ^ 

V partial   product   in    hundreds    order. 

Product,      8399  7  0        in  306  there  are  no  tens  to  be  used 
as  a  multiplier. 

(2)  (3)  (4)  (5)  (6)  (7) 

4086       32607        7908       8099       60772       86507 
4008  4009  909       1088  1019  9003 


8.  Enos  lived  905  years :  how  many  days  did  he 
live,  allowing   365    days   to   the   year? 

9.  A  planter  raised  208  bales  of  cotton,  each  bale 
weighing  475  pounds:  how  many  pounds  of  cotton 
did    he   raise? 

10.  If  a  garrison  of  soldiers  consume  4865  pounds 
of  bread  a  day,  how  many  pounds  will  supply  the 
garrison  408  days?     606  days? 

11.  What  will   508   horses    cost  at   $125   apiece? 

12.  What  will  it  cost  to  build  705  miles  of  railroad 
at  $7525  a  mile? 

LESSON    V. 

JVe^f^  MulttpUcand  J^lffiirey  8, 

MENTAL   EXERCISES. 

1.  Three  times  8  are  how  many?  5  times  8?  7 
times  8?     9  times  8?     8  times  8? 

2.  How  many  are  5  times  8?  8  times  5?  6  times 
8?     8  times  6?     7  times  8?     8  times  7? 


MULTIPLICATION.  55 

3.  There  are  8  quarts  in  1  peck :    how  many  quarts 
in  3  pecks?     5  pecks?     7  pecks? 

4.  There  are  8  pints  in  a  gallon :    how  many  pints 
in  4  gallons?     6  gallons?     8  gallons? 

5.  If  5    men   can   mow  a  field  of  grass  in  8  days, 
how  long  would  it  take  1  man  to  do  it? 

6.  If  a    quantity   of  provisions   will   last   7    men    8 
days,  how  long  will  it  last  1  man? 

7.  If  4  equal  pipes  will  empty  a  cistern  in  8  hours, 
how  long  will  it  take  1  pipe  to  empty  it? 

8.  If  a  tnan  earn  8  dollars  a  week,  how  much  will 
he  earn  in  9  weeks?     11  weeks? 

9.  A  railroad  car  has  8  w^heels:    how  many  wheels 
has  a  train  of  7  cars?     9  cars? 

.     10.  If  a  horse  eat  8  quarts  of  oats  each  day,  how 
many  will  he  eat  in  6  days?     10  days? 

11.  If  a  pint  of  oil  cost  8  cents,  what  will   8  pints 
cost? 

12.  James   has    8   marbles,   and   John    has    6    times 
as  many :    how  many  marbles  has  John  ? 

13.  What  will   8   pounds   of  beef  cost   at   10    cents 
a  pound? 

J^rtfUtpllcand  6>r  Jlfnltiplle?*  endhig  with  Ciphers. 
WRITTEN   EXERCISES. 

1.  Multiply  148000  by  47. 

PROCESS.  rpo   shorten   the  process,  write   the 

14  8000  multiplier   under  the  significant   fig- 

47  ures  of  the  multiplicand,  and,  omit- 

Partial      (1036  ting  the  ciphers  in  forming  the  par- 

products,  I  592  ^^^^    products,    annex    them    to    the 

Product]     7956000  P"^^"^^    obtained.      The    result   will 

be  the  true   product. 


56  INTERMEDIATE  ARITHMETIC. 

Note. — The  teacher  should  sliow  that  the  use  of  the  ciphers 
in  forming  the  partial  products  would  produce  the  same  result. 


(2) 

(3) 

(4) 

(5) 

48000 

308000 

295 

4306 

36 

405 

43000 

245000 

6.  There  are  5280  feet  in  a  mile :  how  many  feet 
in  805  miles? 

7.  The  earth  moyes  in  its  orbit  at  an  average  rate 
of  68000  miles  in  an  hour:  how  far  does  it  move 
in  24  hours?     In  48  hours?  ^ 

8.  If  a  carriage-wheel  revolve  280  times  in  running 
a  mile,  how  many  times  will  it  revolve  in  running 
68  miles?     75  miles? 

9.  A  canal -boat  was  loaded  with  245  bales  of  hay, 
weighing  280  pounds  each:  what  was  the  weight  of* 
the  cargo? 

10.  There  are  480  sheets  of  paper  in  a  ream :  how 
many  sheets  are  there  in  604  reams? 

11.  If  an  acre  of  land  produce  380  pounds  of  cot- 
ton, how  many  pounds  will  248  acres  produce? 

12.  A  steam-boat  makes  145  trips  in  a  season,  and 
carries,  on  an  average,  280  passengers  each  trip  :  how 
many  passengers  does  she  carry  during  the  season? 


LESSON    VI. 

Ji^cH^   MiiUipl'ica7id  I^l(5iire,    9. 
MENTAL  EXERCISES. 

1.  Three   times  9   are  how  many?     4   times   9?     6 
times  9?     8  times  9?     7  times  9?     9  times  9? 

2.  How  many  are  5  times  9?     9  times  5?     7  times 
9?     9  times  7?     10  times  9?     9  times  10? 


MULTIPLICATION.  57 

3.  How  many  are  5  times  10?  10  times  5?  7  times 
10?     10  times  7?     9  times  10?     10  times  9? 

4.  A  man  gave  7  boys  9  rabbits  each  :  how  many 
rabbits  did  he  give  them  all? 

5.  If  a  man  earn  10  dollars  a  week,  how  much 
will  he  earn  in  8  weeks? 

6.  Jane  writes  9  lines  each  day  at  school :  how 
many  lines  does  she  write  in  8  days? 

7.  Charles  receives  9  dollars  a  month  as  errand- 
boy:    how  much  will  he  earn  in  10  months? 

8.  If  7  Inen  can  do  a  piece  of  work  in  9  days, 
how  many  men  will  it  take  to  do  the  same  work 
in  one  day? 

9.  If  a  quantity  of  provisions  will  supply  10  men 
9  days,  how  long  will  it  supply  one  man? 

10.  What  will  6  barrels  of  flour  cost,  at  $9  a  barrel? 


^oth  Multfj^licand  cmd  Mnlfipller  ending 
^WRITTEN    EXERCISES. 

1.  Multiply  198000  by  8900. 

PROCESS.  Write  the  significant   figures  of 

198000  t^^^  multiplier  under  the  significant 

8900  figures    of   the    multiplicand,    and 

^rjoiy  multiply,   omitting   the    ciphers    in 

^^o4  forming   the    partial    products,   but 

annexina:  them  to  the  product  oh- 

Product,  1762200000     ,^.^^^  ^^^  ,^^  ,^^^^  p,,^^,, 

(2)  (3)  (4)  (5) 

94000     90800       470000     950000 
1600       370000     1900       360000 


58  INTERMEDIATE   ARITHMETIC. 

6.  There  are  3600  seconds  in  one  hour :  how  many 
seconds  are  there  in  630  hours? 

7.  Light  moves  192000  miles  in  a  second:  how  far 
does  it  move  in  one  hour? 

8.  A  ship  has  provisions  enough  to  allow  the  crew 
130  pounds  a  day  for  90  days :  how  many  pounds 
of  provisions  are  aboard? 

9.  What  will  1700  tons  of  railroad  iron  cost  at 
$250  a  ton? 

10.  An  army  is  composed  of  54  regiments,  contain- 
ing, on  an  average,  670  men  each :  how  many  men 
in  the  army? 

11.  If  a  steamer  can  run  260  miles  a  day,  how 
far  can  it  run  in  10  days?     In  100  days? 

12.  In  a  field  of  corn  there  are  70  rows,  and  each 
row  contains  280  hills,  and  each  hill  3  stalks:  how 
many  stalks  of  corn  in  the  field? 


LESSON    VII. 

1.  What  will  4  bananas  cost  at  5  cents  apiece? 

2.  What  will  5  barrels  of  flour  cost  at  $9  a  barrel  ? 

3.  If  an  orange  is  worth  5  apples,  how  many  apples 
are  7  oranges  worth? 

4.  If  there  are  8  pints  in  a  gallon,  how  many  pints 
are  there  in  6  gallons? 

5.  Two  men  start  from  the  same  place,  and  travel 
in  opposite  directions,  one  at  the  rate  of  3  miles  an 
hour  and  the  other  4  miles  an  hour:  how  far  will 
they  be  apart  in  8  hours? 

6.  If  an  orange  is  worth  2  lemons  and  a  lemon  is 
worth  5  plums,  how  many  plums  are  worth  6  oranges? 


MULTIPLICATION.  59 

7.  If  7  men  can  do  a  piece  of  work  in  5  days,  how 
long  would  it  take  1  man  to  do  it? 

8.  If  6  men  can  cut  a  field  of  grass  in  8  days, 
liow  many  men  will  it  take  to  cut  it  in  1  day? 

9.  If  3  pipes  fill  a  cistern  in  10  hours,  in  how 
many  hours  will  1  pipe  fill  it? 

WRITTEN   EXERCISES. 

1.  What  is  the  product  of  4894  X  37? 

2.  What  is  the  product  of  5680  X  340? 

3.  6084  X  3008  =  how  many  ? 

4.  704000  X  4800  =  how  many? 

5.  Multiply  forty-eight  thousand  by  sixty -five  thou- 
sand. 

6.  In  a  train  of  37  cars,  each  car  contains  9850 
pounds  of  freight :    how  much  freight  in  the  train  ? 

7.  If  980  pounds  of  bread  will  supply  the  inmates 
of  the  State  Prison  one  day,  how  many  pounds  will 
supply  them  one  year,  or  365  days? 

8.  If  a  sack  of  salt  contain  168  pounds,  what  will 
be  the  weight  of  1600  sacks? 

9.  A  merchant  bought  18  firkins  of  butter,  each 
weighing  32  pounds,  at  37  cents  a  pound :  what  did 
it  cost? 

10.  A  train  of  27  cars  is  loaded  with  iron;  each  car 
contains  48  bars,  and  each  bar  weighs  365  pounds: 
what  is  the  weiirht  of  the  car^o? 


DEPINITIONS,  PRINCIPLES,  AND  EULE. 

Art.  33.  MiilUpllcation  is  the  process  of  taking 
one  number  as  many  times  as  there  are  units  in 
another.     (See  Multiplication  Table,  p.  213.) 


60  INTERMEDIATE   ARITHMETIC. 

The  Multiplicand  is  the  number  taken  or  mul- 
tiplied. 

The  MaltipUer  is  the  number  denoting  how  many 
times  the  multiplicand  is  taken. 

The  Product  is  the  number  obtained  by  multi- 
plying. 

The  multiplicand  and  multiplier  are  called  the 
Factors   of  the    product. 

Art.  34.  The  Sign  of  Multiplication  is  X  ,  and  is 
read  multiplied  by.  When  placed  between  two  num- 
bers, it  shows  that  the  number  before  it  is  to  bo 
multiplied  by  the  number  after  it.  Thus :  6  X  3  is 
read  6  multiplied  by  3. 

Note. — Since  a  change  in  the  order  of  the  factors  does  not  change 
tlie  product,  6X3  may  also  be  read  6  times  3. 

Art.  35.  Multiplication  is  a  short  method  of  addition. 

The  sum  of  5  +  5  -[-  ^  -|-  5  is  the  same  as  4  times  5. 

Art.  36.  Rule  for  Multiplication.  —  1.  Write  the 
multiplier  under  the  mult iplic and ^  jplachig  units  iinder 
units^  tens  under  tens,  etc. 

2.  When  the  multiplier  consists  of  but  one  term,  begin 
at  the  right  and  multiply  successively  each  term  of  the 
multiplicand,  ivriting  the  right-hand  term  of  each  residt 
in  the  product  and  adding  the  left-hand  term  to  the 
next  result. 

3.  When  the  multiplier  consists  of  more  than  one  term, 
multiply  the  multiplicand  successively  by  each  significant 
term  of  the  multiplier,  writing  the  first  term  of  each 
partial  product  under  the  term  of  the  multiplier  which 
produces  it. 

4.  Add  the  particd  products  thus  obtained^  and  the 
sum  will  be  the  true  product. 


MULTIPLICATION  61 

Art.  37.  1.  When  the  multiplier  or  multiplicand,  or 
both,  end  with  one  or  more  ciphers,  omit  the  ciphers 
in  the  partial  products  and  annex  them  to  the  product 
obtained. 

2.  Any  number  may  be  multiplied  by  10,  100,  1000, 
etc.,  by  annexing  to  it  as  mayiy  ciphers  as  there  are 
ciphers  in  the  multiplier. 

LESSON    VIII. 

Problems  co?nbhim^  cidditlo7i,  Stibtractiojr,  and 
Mulitplication . 

MENTAL    EXERCISES 

1.  6x7  +  4  +  5  +  8  +  7  —  6==  how  many? 

2.  8X4 +6  —  3  +  2  —  5  +  6==  how  many? 

3.  A  grocer  bought  10  barrels  of  apples,  at  %\  a 
barrel,  and  sold  them  so  as  to  gain  $15 :  for  how 
much  did  he  sell  them? 

4.  John  has  6  marbles,  and  Willis  has  4  times  as 
many  less  9,  and  Charles  has  as  many  as  both  John 
and  W^illis:    how  many  marbles  has  Charles? 

5.  A  lady  teacher  receives  $9  a  week,  and  spends 
S6  for  board  and  washing :  how  much  can  she  save 
in  8  weeks? 

6.  Two  men  start  from  the  same  place  and  travel 
in  opposite  directions,  one  at  7  miles  an  hour  and  the 
other  at  5  miles  an  hour :  how  far  will  they  be  ap>art 
in  8  hours? 

7.  Two  stages  start  from  the  same  place  and  go  in 
the  same  direction,  one  at  9  miles  an  hour  and  the 
other  at  6  miles  an  hour:  how  far  w^ill  they  be  apart 
in  5  hours? 

8.  When   oranges   are   sold   at  7   cents   apiece   and 


C2  intp:rmediate  arithmetic, 

lemons   at   5   cents   apiece,  how  many  cents  will  buy 
6  oranges  and  8  lemons? 

9.  If  a  man  earn  $8  a  week  and  a  boy  $3,  how 
much  will  they  both  earn  in  7  weeks? 

10.  A  pedestrian  left  a  city  and  walked  9  hours  at 
the  rate  of  4  miles  an  hour ;  he  then  returned  at  the 
rate  of  3  miles  an  hour,  but  in  4  hours  stopped  to 
rest:    how  far  was  he  from  the  city? 

11.  If  a  man  earn  $12  a  week  and  82)end  $7,  how 
much  will  he  save  in  9  weeks? 

WRITTEN  EXERCISESo 

1.  From  4080  X  26  take  2024  X  16. 

2.  A  grocer  bought  275  barrels  of  flour  for  $2475, 
and  sold  it  at  $12  a  barrel:    what  did  he  gain?   . 

3.  A  clerk  receives  $125  a  month,  and  spends  $68 
a  month:    how  much  does  he  lay  up  each  year? 

4.  An  agent  sold  48  sets  of  outline  maps,  at  $16  a 
set ;  the  maps  cost  him  $10  a  set :  how  much  did  he 
make  ? 

5.  If  a  steamer  carry,  on  an  average,  75  passengers 
each  trip,  how  many  passengers  will  it  carry  in  12 
weeks,  making  3  trips  a  week  ? 

6.  A  book  contains  288  pages,  each  page  contains 
42  lines,  and  each  line  13  words:  how  many  words 
in  the  book? 

7.  A  man  bought  a  farm  for  $4780;  he  sold  80 
acres  at  $33  an  acre,  and  the  remaining  portion  for 
$2560 :    how  much  did  he  make  by  the  transaction  ? 

8.  A  regiment  contains  960  men,  exclusive  of  the 
commissioned  officers ;  the  men  receive  $16  a  month, 
and  the  aggregate  salary  of  the  officers  is  $2800  a 
month:    what  is  the  monthly  pay  of  the  regiment? 

9.  A  drover  bought  480  head  of  cattle  in  Ohio,  at 


DIVISION.  63 

$45  a  head,  shipped  them  to  New  York,  at  an  ex- 
pense of  $6  a  head,  and  then  sold  them  at  $56  a 
head:   how  much  did  he  make? 

10.  A  miller  manufactured  560  barrels  of  flour,  and 
sold  it  at  $9  a  barrel ;  the  wheat  cost  $2750,  and  the 
expense  of  running  the  mill  was  $960:  how  much 
did  he  make? 

11.  A  man  sold  5  horses  at  $87  apiece,  and  received 
$350  in  cash  and  a  note  for  the  balance :  what  was 
the  value  of  the  note  ? 

12.  The  President's  salary  is  $50000  a  year:  if  his 
expenses  are  $2500  a  month,  how  much  can  he  save 
during  his  term  of  4  years? 

13.  If  a  quantity  of  provisions  will  supply  960 
soldiers  27  days,  how  many  soldiers  will  it  supply 
one  day? 


SECTION    Y. 

Divisiojy. 


LESSON    I. 

1.  How  many  times  is  2  contained  in  6?     2  in  12? 
•2  in  16?     2  in  18?     2  in  20? 

2.  How  many  times  is  3  contained  in  9?     3  in  12? 
3  in  15?     3  in  18?     3  in  21  ?     3  in  27? 

3.  How  many  times  is  2  contained  in  12?     3  in  12? 
2  in  18?     3  in  18?     2  in  24?     3  in  24? 

4.  Two  boys  sit  at  1   desk :    how  many  desks  will 
seat  8  boys?     16  boys? 


64  INTERMEDIATE  ARITHMETIC. 

5.  If  a  man  walk  3  miles  an  hour,  how  long  will 
it  take  him  to  walk  15  miles?     21  miles? 

Solution. — At  3  miles  an  hour,  it  will  take  as  many  hours  to 
walk  15  miles  as  3  miles  are  contained  times  in  15  miles,  which 
are  5:  it  will  take  5  hours. 


^WRITTEN   EXERCISES. 

I.  Divide  848  by  2. 

PROCESS.  Write   the   divisor  at   the   left 

T^.  .        ^^  r.  .  r.    ^.  .  y     ,       ^f*  thc    dividcnd,    and    draw    a 

JjLVisor,  2)848,  Dividends  j    t        i.  4.  ^1,  ^ 

a^Ta    n      ■  curved    hne   between    them,  and 

^  ^  *      a  straight    line    under   the   divi- 

dend. Begin  at  the  left,  and 
divide  successively  each  term  of  the  dividend  by  the  divisor. 
The  quotient  is  424. 

(2)  (3)  (4)  (5)  (6) 

2)482         2)8642         3)6936        3)9369        3)3696 

7.  Divide  3609  by  3.     8084  by  2. 

8.  Divide  4684  by  2.     6309  by  3. 

9.  At  $3  a  bushel,  how  many  bushels  of  wheat  can 
be  bought  for  $963?     For  $639? 

10.  In  how  many  hours  can  a  man  walk  396  miles, 
if  he  w^alk  at  the  rate  of  3  miles  an  hour? 

II.  If  a  man  earn  $2  a  day,  how  long  will  it  take 
him  to  earn  $360? 

LESSON    II. 

Jl^cH'  DIpIso7^   I^i^u7^es,   ^  and  S. 

MENTAL  EXERCISES. 

1.  How  many  times  is  4  contained  in  8?     4  in  12? 
4  in  20?     4  in  28?     4  in  36? 


DIVISION.  65 

2.  How  many  5's  in  15?  30?  40?  25?  50?  35? 
45?     20? 

3.  Ill  an  orchard  there  arc  16  trees,  in  rows  of  4 
trees  each :   how  many  rows  in  the  orchard  ? 

4.  How  many  ranks  of  4  soldiers  each  will  24  sol- 
diers make?     32  soldiers?     40  soldiers? 

5.  A  man  planted  30  peach  trees  in  rows,  setting 
5  trees  in  each  row :  how  many  rows  did  they  make  ? 

6.  A  school -room  contains  35  desks,  arranged  with 
5  desks  in  each  row:  how  many  rows  of  desks  in  the 
room  ? 

7.  How  many  chairs,  at  $4  apiece,  can  be  bought 
for  $36? 

8.  How  many  pairs  of  boots,  at  $5  a  pair,  can  be 
bought  for  $35? 

0.  Mary  is  reading  5  chapters  a  day :  how  long  will 
it  take  her  to  read  45  chapters? 

10,  A  boy  had  50  peach -stones,  which  he  jilanted 
in  rows  of  5  each:    how  many  rows  did  he  plant? 

A^TKITTEN  EXEKCISES. 

1.  Divide  784  by  4. 

PROCESS.  Write  the  divisor  at  the  left 

r..  .         ,s^r>       T^    .  T     T         of  the  dividend.     Begin  at  the 
Devisor,  4 ) 784,  Dwidend,        i<,ft.,,,^„,,  ^^^  „f  the  dividend, 

196,    Quotient.  ^nd  divide,  thus;  4  is  contained 

in  7  hundreds  1  hundred  times, 
with  3  hundreds  remaining.  Write  the  1  hundred  in  hun- 
dreds' place  in  the  quotient,  and  reduce  the  3  hundreds  re- 
maining to  30  tens,  which,  with  the  8  tens  added,  make  38 
tens.  Four  is  contained  in  88  tens  9  ten  times,  with  2  tens 
remaining.  Write  the  9  tens  in  tens'  place  in  the  quotient, 
and  reduce  the  2  tens  remaining  to  20  units,  which,  with. the 
4  units  added,  make  24  units.  Four  is  contained  in  24  units 
I.  A.— 5. 


66  INTERMEDIATE  ARITHMETIC. 

6  times.     Write  the  6  units  in  units'  place  in  the  quotient. 
The  quotient  is  196. 

(2)  (3)  (4)  (5)  (G) 

4)764  4)936  5)640  5)870  5)765 

7.  Divide  1128  by  2 ;   by  3 ;   by  4. 

8.  Divide  8740  by  2 ;    by  4 ;    by  5. 

9.  Divide  18480  by  2;   by  3;    by  4;    by  5. 

10.  A  mMnufacturer  packed  372  clocks  in  boxes, 
placing  4  clocks  in  each  box:  how  many  boxes  were 
required  ? 

11.  If  4  bushels  of  wheat  will  make  a  barrel  of 
flour,  how  many  barrels  will  972  bushels  make? 

12.  If  a  man  earn  $4  a  day,  how  many  days  will 
it  take  him  to  earn  $1584? 

LESSON    III, 

A^cw  divisor   J^/'^urcs^   6  and  7. 

MENTAL  EXEKCISES. 

1.  Six  is  contained  in  12  how  many  times?  6  in 
24?     6  in  36?     6  in  48?     6  in  54? 

2.  How  many  times    7    in    7?     7   in    21?     7   in   35? 

7  in  49?     7  in  63?     7  in  42?     7  in  56? 

3.  How  many  6's  in  42?  7's  in  42?  6's  in  30? 
5's  in  30?  7's  in  28?  4's  in  28?  7's  in  35?  5's  in 
35?     7's  in  56? 

4.  If  6  chairs  make  a  set,  how  many  sets  will  36 
chairs  make?     48  chairs?     60  chairs? 

5.  There  are  7  da^^s  in  a  week :  how  many  weeks 
in  49  days?     In  56  days?     63  days? 

6.  There  are  6  feet  in  a  fathom :  how  many  fathoms 
in  54  feet?     In  60  feet? 


DIVISION.  67 

7.  An  orchard  contains  56  trees  in  rows  of  7  trees 
each:    how  many  rows  of  trees  in  the  orchard? 

8.  How  many  plows,  at  $6  each,  can  be  bought  for 
$48?     For  $54? 

WRITTEN  EXERCISES. 

4)784(190 
Solve  each  of  the  written  exercises  in  the  4 

preceding  lesson  (II),  writing  the  quotient  at  r^ 

the   right  of    the  dividend,   and    the   partial  o/> 

dividends  and  products  below  the  dividend, 
as  at  the  right: 


2  4 
24 


1.  Divide  4876  by  23, 


PROCESS.  Divide  48  hundreds  by  23,  and 

23)4876(212,  quotient.        write  the  result,  2  hundreds,  at 

46  the  right  of  the  dividend  for  the 

~^  first   or    hundreds'   term  of  the 

quotient.     Multiply  the   divisor 

(23)  by  this  quotient  term,  and 

'*"  subtract  the    product,    46    huur 

^^  dreds,   from    48   hundreds.      To 

the  remainder,  2  hundreds,  an- 
nex the  7  tens  of  the  dividend,  giving  27  tens  for  the  second 
partial  dividend.  Divide  27  tens  by  23,  and  write  the  result, 
1  ten,  for  the  tens'  term  of  the  quotient.  Multiply  23  by 
this  1  ten,  and  subtract  the  product,  23  tens,  from  27  tens. 
To  the  remainder,  4  tens,  annex  the  6  units  of  the  dividend, 
giving  46  units  for  the  third  partial  dividend.  Divide  46 
units  by  23,  and  write  the  result,  2,  for  the  units'  term  of 
the  quotient.  Multiply  23  by  2,  and  subtract  the  result 
from  46.     The  quotient  is  212. 

Note. — In  tliis  and  the  next  eight  problems,  each  term  of  the 
quotient  may  be  determined  hy  dividing  the  left-hand  term  of  the 
partial  dividend  bi/  the  left-hand  term  of  the  divisor.  In  the  last  five 
problems  (6  to  10,  inchisive)  each  term  of  the  quotient  may  also 
be  determined  by  dividing;  the  two  left-hand  terms  of  the  partial 
dividend  by  the  two  left-hand  terms  of  the  divisor. 


68  INTERMEDIATE  ARITHMETIC. 

2.  Divide  4686  by  22.     68952  by  221. 

3.  Divide  3813  by  123.     63336  by  203. 

4.  Divide  446886  by  2013. 

5.  Divide  678273  by  2113. 

6.  Divide  549661  by  1043. 

7.  Divide  69898188  by  10678. 

8.  Divide  4890375  by  1035. 

9.  Divide  35854660  by  10435. 

10.  Divide  45691212  by  10562. 

11.  Divide  1608  by  67. 


PROCESS. 

67)1608(24,  Quotient 
134 

268 
268 


Since  67  is  not  contained  in  the 
number  denoted  by  the  first  two 
left-hand  terms  of  the  dividend, 
take  160  tens  for  the  first  partial 
dividend. 


Note. — It  is  sometimes  difficult  to  tell  how  many  times  the 
divisor  is  contained  in  a  partial  dividend.  Wlien  this  is  the  case, 
take  the  first  left-hand  term,  or  first  two  left-hand  terms,  of  the 
divisor  for  a  trial  divisor,  and  the  proper  number  of  left-hand  terms 
of  the  partial  dividend  for  a  trial  dividend. 

If  the  divisor,  multiplied  by  the  quotient  term  thus  found,  gives 
a  product  greater  than  the  partial  dividend,  tlie  quotient  term  is  too 
large,  and  should  be  reduced.  Tiie  trial  divisor  in  the  above  ex- 
ample is  6,  the  first  trial  dividend  is  16,  and  the  second  26.  The 
teacher  should  make  this  process  plain  to  tlie  pupil. 


12.  Divide  312  by  24.     374  by  17. 

13.  Divide  792  by  36.     1625  by  65. 

14.  Divide  2520  by  36.     3024  by  63. 

15.  Divide  64347  by  267.     49179  by  507. 

16.  There    are    36    inches    in    a    yard :    how    many 
yards  are  there  in  792  inches? 

17.  A  bushel  of  corn  weighs  56  pounds :  how  many 
bushels  of  corn  in  24416  pounds? 


DIVISION.  69 

18.  A   hogshead    of  molasses    contains    63   gallons : 
how  many  hogsheads  in  4788  gallons? 

19.  If  72  books  can  be  packed  in  a  box,  how  many 
boxes  will  it  take  to  hold  17496  books? 

20.  How  many  farms  of  156  acres  each  can  be  sold 
from  a  tract  of  land  containing  7332  acres? 

21.  If  a  vessel  sail,  on  an  average,  47  miles  a  day, 
how  long  will  it  take  it  to  sail  2303  miles? 

22.  There    are    365    days   in   a  common  year :    how 
many  years  are  there  in  90155  days? 


LESSON    IV. 

JVeH^  divisor  I^igiire,  8, 

MENTAL  EXERCISES. 

1.  How  many  times  is  8  contained  in  8?     8  in  24? 
8  in  40?     8  in  56?     8  in  72? 

2.  How   many   8's   in    56?     7's   in   56?     S's   in   48? 
6's  in  48?     8'8  in  72?     9's  in  72? 

3.  32   --   8   ==   how    many?      49    -r-   7?      54  -f-   6? 
64  ~  8?      56-4-7?      56  --  8? 

4.  There  are  8  quarts  in  a  peck :   how  many  pecks 
in  72  quarts? 

5.  If  a  steamer  run  8  miles  an  hour,  in  how  many 
hours  will  it  run  80  miles? 

6.  There  are  8  furlongs  in  a  mile:  how  many  miles 
in  56  furlongs? 

7.  At  $8  a  barrel,  how  many  barrels   of  flour  can 
be  bought  for  $64? 

8.  If  a    man    work    8    hours   a   day,   in   how  many 
days  will  he  work  96  hours? 


70  INTERMEDIATE  APaXHMETIC. 

The  Quotient  contahi'mg  One  or  more  Cip/iers, 
WRITTEN  EXERCISES. 

1.  Divide  341:^7  by  84. 

PROCESS.  Since    the   divisor    is   not    con- 

84)34137(406  tained  in  the  second   partial  div- 

33g  idend   (53   tens),   Avrite    0    in    the 

yz^  tens^   phice   in   the   quotient,   and 

_  ^  .  annex  the  7  units  for  a  third  par- 
»)  04 

tial  dividend.     As  there  is  no  hg- 

3  3,  Remainder.       ^^^  ^^f  ^j^^  dividend  left  to  annex 

to  33  to  form  a  new  partial   divi- 
dend, 33  remains  undivided,  and  is  called  the  remainder. 

2.  Divide  24399  by  48.     4G7034  by  806. 

3.  Divide  2845007  by  5728.     215607  by  18036. 

4.  Divide  1423685  by  6785.     1604083  by  2088. 

5.  In  1  week  there  are  168  hours:  how  many 
weeks  in  85008  hours? 

6.  A  drover  went  West  with  $23490  to  buy  cattle: 
how  many  cattle  could  he  buy  at  $58  a  head  ? 

7.  If  a  garrison  consume  648  pounds  of  bread  in  a 
day,  how  long  will  134136  j)ounds  last  it? 

8.  If  the  average  daily  receipts  of  a  ferry-boat  are 
$275,  in  how  many  days  will  the  receipts  amount  to 
$165825? 

9.  How  long  will  it  take  a  pipe,  discharging  158 
gallons  of  water  in  an  hour,  to  empty  a  cistern  con- 
taining 7584  gallons? 

10.  A  cord  of  wood  contains  128  solid  feet:  how 
many  cords  in  a  pile  containing  5280  solid  feet? 

11.  Divide  543513392  by  6704. 


DIVISION.  71 


^  LESSON   V. 

MENTAL  EXERCISES. 

1.  How  many  times  9  in  18?     9   in  27?     9  in  36? 
9  in  54?     9  in  72?     9  in  90? 

2.  How  many  9's   in   45?     5's   in    45?     9's   in    63? 
7's  in  63?     9'8  in  72?     8's  in  72? 

3.  How  long  will  it  take  a  steamer  to  make  a  trip 
of  81  miles  if  it  run  9  miles  an  hour? 

4.  If  9    words   fill   a   line,  how  many  lines  will  72 
words  fill?     81  words? 

5.  If  a   man   can   do   a   piece  of  work  in  90  days, 
how  many  men  can  do  it  in  9  dnys? 

6.  If  a  quantity  of  provisions  will  last  72  men  one 
day,  how  long  will  it  last  9  men? 

7.  How  many  sheep,  at   $9    a   head,  can  be  bought 
for  $54?     For  $63? 

8.  A  copy-book  contains  100  lines,  with  10  lines  on 
each  page:    how  many  pages  in  the  book? 

9.  If  a  man  earn  $10  a  week,  how  long  will  it  take 
him  to  earn  $100? 

10.  How  many  tons  of  hay,  at  $10   a   ton,  can  be 
bought  for  $90? 

11.  8x6^8  +  9  —  7--  what? 

12.  9x4---6  +  7-;|-6  —  8==  what? 

13.  72  --  9  X  6  -f-  8'x  3  --  9  =  what? 

14.  81-f-9--3x8--Gx8  —  9  =  what? 

15.  7x8-^7x9  —  9  +  4  —  8::=  what? 

To  THE  Teacher. — For  additional  examples  see  Manual 
OF  Arithmetic,  page  39. 


72 


INTERMEDIATE   ARITHMETIC. 


The  divisor  ending  hi  One  07*  more  Cfp/ie7^s» 


WRITTEN  EXERCISES. 


1.  Divide  350  by  10. 


FIRST  PROCESS.  By  comparing  these  two  pro- 

cesses, it  is  seen  that  350  is 
divided  by  10,  by  cutting  off 
the  right-hand  figure.  Tlie 
reason  is  obvious.  Tlie  cut- 
ting off  of  the  right-hand  fig- 
ure removes  each  of  the  other 
figures  one  place  to  the  right, 
and  thus  decreases  their  value 
ten-fold.  In  like  manner,  it 
may  be  shown  that  cutting 
off  the  two  right-hand  figures 
divides  a  number  by  100; 
cutting  off  three  right-hand 
65,  Remainder,    figures,  by  1000,  etc. 


10)350(35 
30 

50 
50 

SECOND  PROCESS. 

1 1  0 )35|0 

3  5,  Quotient. 

2.  Divide  2865  by  100 
1|00)28|65 

2  8,         Quotient. 


3.  Divide  45600  by  10.     By  100. 

4.  Divide  187000  by  1000.     By  100. 

5.  Divide  384050  by  100.     By  1000. 

6.  Divide  230045  by  1000      By  10000. 

7.  Divide  450860  by  10000.     By  1000. 

8.  Divide  196800  by  4800. 


PROCESS. 
48100)1968100(41,  Quotient. 
192 

48 
48 


First  divide  both  divisor 
and  dividend  by  100,  which 
is  done  by  cutting  off  the 
two  right-hand  figures. 
Then  divide  1968,  the  new 
dividend,  by  48,  the  new 
divisor.     The  quotient  is  41. 


Note.— The  teacher  can  show  that  both  divisor  and  dividend  may  he 
divided  by  any  number  withont  affecting  the  value  of  the  quotient. 


PROCESS. 

45100)588164(13 
45 
138 
135 

3  64,  Remainder. 

DIVISION.  73 

9.  Divide  63200  by  7900. 

10.  Divide  116087000  by  2900. 

11.  Divide  70338000  by  75000. 

12.  Divide  58864  by  4500. 

First  divide  both  divisor  and 
dividend  by  100,  wliicli,  in  the 
case  of  the  dividend,  leaves  a 
remainder  of  64.  Next  divide 
588  by  45,  leaving  a  remainder 
of  3  hundreds,  to  wliich  annex 
the  first  remainder  (64),  obtain- 
ing 364  for  the  true  remainder. 

Note. — Tlie  trne  remainder  is  found  by  annexing  the  first  remain- 
der to  the  second.  The  reason  for  this  can  be  easily  given  by  the 
teacher. 

13.  Divide  466384  by  3900.     220345  by  940. 

14.  Divide  99990  by  5400.     172800  by  14400. 

15.  A  barrel  of  beef  contains  200  pounds:  how 
many  barrels  will  contain  12800  pounds? 

16.  There  are  480  sheets  of  paper  in  a  ream :  how 
many  reams  will  129600  sheets  make? 

17.  There  are  3600  seconds  in  an  hour:  how  many 
hours  in  172800  seconds? 

18.  How  many  city  lots,  at  $1600  each,  can  bo 
bought  for  $25600? 

19.  How  many  cars,  each  carrying  1800  pounds, 
will  transport  79200  pounds  of  hay? 

20.  How  many  barrels,  each  holding  196  pounds, 
will  hold  9016  pounds  of  flour? 

21.  How  many  regiments,  averaging  750  men  each, 
will  make  an  army  of  30000  men  ? 

22.  A  peach  orchard  contains  6758  trees,  and  there 
are,  on  an  average,  62  trees  on  each  acre :  how  many 
acres  in  the  orchard? 


T4  INTERMEDIATE  ARITHMETIC. 

23.  A  pipe  discharges  94  gallons  in  an  hour:  in 
how  many  iioiirs  will  it  empty  a  cistern  holding  3384 
gallons  of  water? 

24.  What  number  multiplied  by  98  will  produce 
15288? 

25.  The  dividend  is  5292  and  the  divisor  is  (]3 : 
what  is  the  quotient? 

26.  The  divisor  is  $1500  and  the  dividend  $564000: 
what  is  the  quotient? 

DEFINITIONS,  PEINOIPLES,  AND  EULES. 

Art.  38.  Division  is  the  process  of  finding  how 
many  times  one  number  is  contained  in  another. 

The  Dividend  is  the  number  divided. 

The  Divisor  is  the  number  by  which  the  dividend 
is  divided. 

The  Quotient  is  the  number  of  times  the  divisor 
is  contained  in  the  dividend. 

The  lieniainder  is  the  part  of  the  dividend  which 
is  left  undivided.  When  the  dividend  contains  the 
divisor  an  exact  number  of  times,  there  is  no  re- 
mainder. 

Art.  39.  The  Sign  ^ of  Division  is  ^,  and  is  read 
divided  by.  When  placed  between  two  numbers,  it 
shows  that  the  number  before  it  is  to  be  divided  by 
the  number  after  it.  Thus :  16  -f-  4  =  4  is  read 
16  divided  by  4  equals  4. 

Division  is  also  expressed  by  writing  the  dividend  above 
and  the  divisor  below  a  short  horizontal  line.  Thus:  \f-  is 
read  18  divided  'by  '6. 

Art.  40.  One  number  is  contained  in  another  num- 
ber as  many  times  as  it  can  be  taken  from  it.     Hence 


DIVISION.  75 ' 

division  is  a  short  method  of  finding  hgw  many  times 
one  number  may  be  subtracted  from  another. 

A  number  is  contained  in  another  as  many  times 
as  it  must  bo  taken  to  produce  it.  Hence  division 
may  be  regarded  as  the  reverse  of  multiplication. 
The  divisor  and  quotient  are  factors  of  the  dividend. 

Art.  41.  There  are  two  methods  of  division,  called 
Short  Division  and  Long  Division. 

In  Short  Division,  the  partial  products  and  par- 
tial dividends  are  not  written,  but  are  formed  men- 
tally. This  method  is  generally  used  when  the  divisor 
does  not  exceed  12. 

In  Long  Division,  the  j^artial  products  and  partial 
dividends  are  written. 

Art.  42.  EuLE  for  Short  Division.  —  1.  Write  the 
divisor  at  the  left  of  the  dividend^  draw  a  curved  line 
between  them^  and  a  straight  line  under  the  dividend. 

2.  Find  how  many  times  the  divisor  is  contained  in 
the  left-hand  term  or  terms  of  the  dividend^  taken  as  a 
partial  dividend^  and  write  the  quotient  under  the  last 
figure  of  the  dividend  used. 

3.  Midtiply  the  divisor  by  the  quotient  term  found, 
and  subtract  the  product  from  the  partial  dividend  used, 
performing  each  process  mentally. 

4.  Prefix  the  remainder^  if  there  be  one,  to  the  next 
term  of  the  dividend  for  a  second  partial  dividend^  and 
divide,  multiply,  and  subtract,  as  before. 

5.  Proceed  in  this  manner  until  all  the  terms  of  the 
dividend  have  been  used. 

Proof. — Multiply  the  divisor  by  the  quotient,  to  the 
product  add  the  remainder,  if  there  be  any,  and  if  the 
result  equals  the  dividend,  the  work  is  correct. 


76  INTERMEDIATE   ARITHMETIC. 

Art.  43.  EuLE  for  Long  Division.  —  1.  Write  the 
divisor  at  the  left  of  the  dividend,  draw  a  curved  line 
beticeen  them,  and  also  at  the  right  of  the  dividend,  to 
separate  it  from  the  quotient. 

2.  Take  as  many  of  the  left-hand  terms  of  the  dividend 
as  will  contain  the  divisor,  for  a  partial  dividend ;  find 
how  many  times  this  will  contain  the  divisor,  and  write 
the  quotient  at  the  right  of  the  dividend  for  the  first 
left-hand  term  of  the  quotient, 

3.  Multiply  the  divisor  by  the  quotient  term  found, 
write  the  product  under  the  partial  dividend  used,  and 
subtract. 

4.  To  the  remainder  annex  the  next  term  of  the  divi- 
dend for  a  second  partial  dividend,  and  divide,  multiply^ 
and  subtract,  as  before. 

5.  Proceed  in  this  manner  until  all  the  terms  of  the 
dividend  have  been  used. 

Note.— When  any  partial  dividend  does  not  contain  the  divisor, 
write  a  ciplier  in  the  quotient,  and  annex  another  term  of  the  divi- 
dend to  form  a  new  partial  dividend. 

Art.  44.  When  one  or  more  of  the  rigbt-haxid  fig- 
ures of  the  divisor  are  ciphers  — 

1.  Cut  off  the  ciphers  from  the  right  of  the  divisor, 
and  an  equal  number  of  figures  from  the  right  of  the 
dividend. 

2.  Divide  the  yiew  dividend  thus  formed  by  the  new 
divisor,  and  the  result  will  be  the  quotient. 

3.  Prefix  the  remainder,  if  there  be  one,  to  the  figures 
cut  off  from  the  dividend,  and  the  result  will  be  the 
true  remainder. 


DIVISION.  77 

Art.  45.  To  divide  an}^  number  by  10,  100,  1000, 
etc., — 

Cut  off  as  many  figures  from  the  right  as  there  are 
ciphers  in  the  divisor.  The  figures  cut  off  are  the  true 
remainder. 


LESSON    VI. 

MISCBLLA^J^BO  US   "RBriBW  T^CSLBMS. 

1.  The  Bum  of  two  numbers  is  15  and  one  of  the 
numbers  is  6:    what  is  the  other? 

2.  The  difference  between  two  numbers  is  8  and 
the  smaller  number  is  9:    what  is  the  larger? 

3.  The  product  of  two  numbers  is  56  and  one  of 
the  numbers  is  7:   what  is  the  other? 

4.  The  quotient  of  two  numbers  is  6  and  the  di- 
visor is  8:    what  is  the  dividend? 

5.  How  many  barrels  of  flour,  at  $8  a  barrel,  will 
pay  for  24  yards  of  carpeting,  at  $2  a  yard  ? 

6.  How  many  tons  of  coal,  at  $9  a  ton,  will  pay 
for  15  cords  of  wood,  at  $6  a  cord? 

7.  A  grocer  bought  7  barrels  of  flour  at  $6  a  baiTel : 
for  how  much  a  barrel  must  he  sell  it  to  gain  $14 
on  the  lot? 

8.  If  1  man  can  build  a  wall  in  36  days,  how 
many  men  can  build  it  in  4  days? 

9.  If  6  men  can  do  a  piece  of  work  in  8  days, 
how  many  men  can  do  it  in  12  days? 

10.  Two  vessels  start  from  the  same  port  and  sail 
in  the  same  direction,  one  sailing  12  miles  an  hour 
and  the  other  9  miles  an  hour:  how  far  apart  will 
they  be  in  10  hours? 


78  INTERMEDIATE   ARITHMETIC. 


WKITTEN   EXERCISES. 

1.  The  greater  of  two  numbers  is  4056  and  their 
difference  is  3650 :    what  is  the  less  number  ? 

2.  The  subtrahend  is  34203  and  tlie  remainder  is 
8706:    what  is  the  minuend? 

3.  The  divisor  is  534  and  the  quotient  43:  what  is 
the  dividend? 

4.  The  product  of  two  numbers  is  5328  and  one 
of  the  numbers  is  148:    what  is  the  other? 

5.  Multiply  486  +  392  by  their  difference. 

6.  Divide  the  product  of  48  and  24  by  their  differ- 
ence. 

7.  A  merchant  bought  35  yards  of  cloth  at  $56, 
and  sold  it  at  $2  a  yard:    how  much  did  he  gain? 

8.  A  drover  bought  240  sheep  at  $8  a  head,  and 
then  sold  90  of  them  at  $12  a  head,  75  at  $9  a  head, 
and  the  rest  at  $6  a  head:   how  much  did  he  gain? 

9.  A  farmer  exchanges  65  bushels  of  wheat  at  $2 
a  bushel,  and  35  sheep  at  $6  a  head,  for  cows  at  $34 
a  head:    how  many  cows  did  he  receive? 

10.  A  man's  income  is  $3500  a  year;  he  pays  $450 
a  year  for  house-rent,  $150  for  taxes,  $350  for  hired 
help,  and  $45  a  month  for  other  expenses:  how  much 
has  he  left  at  the  close  of  the  year? 

11.  A  man  bought  80  acres  of  land  at  $35  an  acre, 
paid  $325  for  improvements,  and  then  sold  it  for 
$3750:    how  much  did  he  gain? 

12.  A  grain  merchant  having  3500  bushels  of  oats, 
sold  1650  bushels,  and  then  bought  twice  as  much 
as  he  had  left:    how  many  bushels  did  he  buy? 

13.  A  man  left  an  estate  to  his  wife  and  three 
children ;  the  wife  received  $4500 ;  the  youngest 
child,  $1500;  the  second  child,  $1850;   and  the  eldest 


DIVISION.  79 

child,  as  much  as  both  of  the  others  less  $1350:   what 
was  the  value  of  the  estate? 

14.  A  and  B  start  together  on  a  journey,  A  trav- 
eling 28  miles  a  day  and  B  33  miles :  how  far  apart 
will  they  be  in  12  days? 

15.  A  and  B  start  together  and  travel  in  opposite 
dii-ections,  A  going  28  miles  a  day  and  B  33  miles: 
how  far  apart  will  they  be  in  12  days? 


Questions  for  Keview. 

What  is  addition?  What  is  meant  by  sum  or  amount? 
What  does  it  contain?  What  is  meant  by  like  numbers? 
What  numbers  can  be  added  ?  What  is  the  sign  of  addi- 
tion ?  What  does  it  show  ?  Give  the  rule  for  addition. 
What  is  the  method  of  proof? 

What  is  subtraction  ?  The  difference,  or  remainder  ?  The 
minuend?  Tlie  subtrahend?  What  numbers  can  be  sul)- 
tracted?  What  does  the  sum  of  the  remainder  and  subtra- 
hend equal  ?  What  is  the  sign  of  subtraction  ?  What  does 
it  show?  Give  the  rule  for  subtraction.  What  is  a  method 
of  proof? 

What  is  multiplication  ?  The  multiplicand?  The  multi- 
plier? The  product?  Of  what  are  the  multiplicand  and 
multiplier  factors?  What  is  the  sign  of  multiplication? 
What  does  it  show  ?  How  may  the  product  be  obtained  by 
addition  ? 

Give  the  rule  for  multiplication.  How  may  you  multiply 
when  either  the  multiplicand  or  multiplier,  or  both,  end  in 
ciphers?  How  may  any  number  be  multiplied  by  10,  100, 
1000,  etc.? 

What  is  division?  The  dividend?  The  divisor?  The 
quotient?  The  remainder?  What  is  the  sign  of  division? 
What  does  it  show?  In  what  other  way  may  division  be 
expressed  ?  How  many  times  may  the  divisor  be  subtracted 
from  the  dividend?     Of  what  is  division  the  reverse? 


80  INTERMEDIATE   ARITHMETIC. 

What  is  short  division  ?  When  is  it  used  ?  Give  the  rule. 
What  is  long  division  ?  Give  the  rule.  How  do  you  proceed 
when  a  partial  dividend  will  not  contain  the  divisor?  How 
may  you  divide  when  the  divisor  ends  in  ciphers  ?  How 
may  any  number  be  divided  by  10,  100,  1000,  etc.  ? 


SECTIOl^    VI. 


LESSON    I. 
Divisor,  Greatest  Common  Dlyisor,  and  I^actor, 

Note. — The   term   number    used    in    this  section,   denotes    an 
integer. 

1.  What  numbers  besides  itself  and   1   will  exactly 
divide   15?     21?     25?     30?     56?     63? 

2.  What  numbers  besides  itself  and    1   will  exactly 
divide  7?     11?     13?     17?     23?    37?    41? 

3.  What   numbers    will   exactly  divide  4?     5?     16? 
19?     24?     29?     33?     31?     42? 

4.  What   are    the    divisors    of  10?     28?     31?     33? 
43?    49?    53?     55?     70?     90?     99? 

Note. — Since  every  number  is  exactly  divisible  by  itself  and  1, 
these  divisors  need  not  be  given. 

5.  What   number   is   a   divisor   of  both   9   and    12? 
15  and  20?     24  and  27?     42  and  56? 

6.  W^hat  divisor  is  common  to  28  and  35?     27  and 
36?     42  and  54?     63  and  81? 

7.  What   is   a   common   divisor   of  15  and  30?     45 
and  60?     50  and  75?     60  and  84? 


PROPERTIES  OF  NUMBERS.  81 

8.  What  is  the  greatest  divisor  common  to  48  and 
72?     27  and  54?     50  and  75? 

9.  What  is  the  greatest  common  divisor  of  24  and 
36?     32  and  48?     56  and  84? 

10.  What  is  a  common  divisor  of  16,  32,  and  48? 
15,  30,  and  45?     36,  54,  and  72? 

11.  What  is  the  greatest  common  divisor  of  32 
and  48?  15,  30,  and  45?  36,  54,  and  72?  18,  45, 
and  81? 

Art.  46.  A  number  that  has  no  divisor  except  itself 
and  1,  is  called  a  Prime  J\^U7riber.  A  number  that 
has  other  divisors  besides  itself  and  1,  is  called  a 
Composite  Jfumher, 

12.  Name  all  the  prime  numbers  between  0  and  20. 
Between  20  and  30.     40  and  50. 

13.  Name  all  the  composite  numbers  between  20  and 
30.     40  and  50.     60  and  70. 

14.  What  are  the  prime  divisors  of  15?  18?  22? 
28?     33?    36?    37?    40?    43? 

15.  What  are  the  prime  divisors  of  16?  20?  25? 
27?     35?    44?     55?     60? 

Art.  47.  The  divisors  of  a  number  are  called  its 
Factors;  and  prime  divisors  are  called  Prime 
Factors, 

16.  What  are  the  prime  factors  of  21?  24?  35? 
39?    42?    49?     54?     56?     63?     mi    72? 

17.  Of  what  number  are  2,  3,  and  5  prime  factors? 
2,  5,  and  7?     2,  2,  3,  and  5? 

Note. — Tlie  product  of  the  prime  factors  of  a  number  equals 
the  number. 

18.  Of  what  number  are  2,  2,  3,  and  3  prime  factors? 
2,  3,  5,  and  7?     3,  5,  2,  and  7? 

I.  A.— 6. 


82  INTERMEDIATE   ARITHMETIC. 

WRITTEN  EXERCISES. 

1.  What  are  the  prime  factors  of  126? 

PROCESS.  Divide    126    by    2,    a    prime 

2)126  divisor;    next    divide    the  quo- 

g  \  gg  tient  63  by  3,  a  prime  divisor  ; 

oT^T  and    then    divide    the    quotient 

21   by   3,  a  prime  divisor.     The 

prime  factors  are  2,  3,  3,  and  7. 


7 
126  =  2X3X3X7. 


"What  are  the  prime  factors  of 

2.  63?  6.  175?  10.  264?  14.  440? 

3.  72?  7.  147?  11.  200?  15.  500? 

4.  84?  8.  275?  12.  256?  16,  648? 

5.  96?  9.  325?  13.  250?  17.  900? 

18.  What  is   the    greatest   common    divisor   of  126 
and  210? 

PROCESS.  Eesolve  126  and  210  into  their 

126  =  ^X$X3Xyf  prime  factors.     The  product  of 

2io_:,^\/*\/5\/r^  the  factors  common  to  both  will 


2  X  "^  X  7  ==  42,  Alls. 


be  the  greatest  common  divisor 
required. 


Note. — Tliis  process  and  tliat  for  finding  the  least  common 
multiple  (Art.  54)  may  be  easily  explained  by  means  of  objects. 

What  is  the  greatest  common  divisor  of 

19.  54  and  90?  23.  81  and  135? 

20.  72  and  96?  24.  63,  84,  and  126? 

21.  75  and  90?  25.  96,  144,  and  192? 

22.  84  and  108?  26.  128,  224,  and  320? 

Art.  48.  A  Divisor  of  a  number  is  a  number  that 
will  exactly  divide  it. 

A  Common  Divisor  of  two  or  more  numbers  is 
a  number  that  will  exactly  divide  each  of  them. 


PROPERTIES  OP  NUMBERS.  83 

The  Greatest  Common  Divisor  of  two  or  more 
numbers  is  the  greatest  number  that  will  exactly 
divide   each   of  them. 

Art.  49.  A  Prime  Number  is  one  that  has  no 
divisor   except   itself  and    1. 

A  Composite  Number  is  one  that  has  other  di- 
visors  besides   itself  and    1. 

Art.  50.  An  Ei^en  Number  is  exactly  divisible  by 
2;    as,  2,  4,  6,  8,  10,  12,  etc. 

An  Odil  Number  is  not  exactly  divisible  by  2; 
as,  1,  3,  5,  7,  9,  11,  13,  etc. 

All  even  numbers  except  2  are  composite. 

Art.  51.  EuLES.  —  1.  To  resolve  a  composite  num- 
ber into  its  prime  factors.  Divide  it  by  any  "prime  divisor^ 
and  the  quotient  by  any  prime  divisor,  and  so  continue 
until  a  quotient  is  obtained  which  is  a  prime  number. 
The  several  divisors  and  the  last  quotient  are  the  prime 
factors. 

2.  To  find  the  greatest  common  divisor  of  two  or 
more  numbers.  Resolve  the  given  numbers  into  their 
prime  factors^  and  select  the  factors  which  are  common. 
The  product  of  the  common  factors  will  be  the  greatest 
common  divisor. 


LESSON    II. 
Multiple  and  Zeast  Conimo7i  Multiple, 

Art.  52.  When  a  number  is  multiplied  by  an  in- 
teger, the  product  is  called  a  Multiple,  Thus,  3G, 
or  12  X  3,  is  a  multiple  of  12. 

1.  What  number  is  a  multiple  of  3?  4?  5?  7? 
8?     10?     15?     20?     25?     30? 


84  INTERMEDIATE   ARITHMETIC. 

2.  What   number   is    a    multiple    of  18?     24?     35? 
45?     44?     60?     100?     250? 

Note. — The  teacher  should  show  that  a  number  has  any  num- 
ber of  multiples. 

3.  What   number  is   a   common  multiple  of  3  and 
4?    4  and  5?     6  and  8?     5   and  6? 

4.  What  number  is  a  common  multiple  of  7  and  5? 
6  and  9?     3,  4,   and  G?    4,  8,  and  12? 

Note. — The  teacher  should   show  that  two  or  more  numbers 
have  any  number  of  common  multiples. 

5.  What  is  the  least  common  multiple  of  3  and  4? 
5  and   6?     3,  6,  and   12?     2,  4,  and  8? 

6.  What  is  the  least  common  multiple  of  3,  5,  and 
10?    2,  5,  and  10?    2,  3,  5,  and  10? 

WRITTEN   EXEBCISES. 

1.  What   is   the    least   common   multiple  of  12,  18, 
and  30? 

PROCESS.  ^  Resolve  the  num- 

12  =  ^  V  ^  y  3  bers  into  their  prime 

|g__2y*yj^  factors,   and    select 

30  =  2X3X0  all  the  different  fac- 

— tors,  repeating  each 

2x2x3x3x5=r  180,  L.  G,  M.      as  many  times  as  it 

is  found  in  any  num- 
ber. The  factor  2  is  found  twice  in'  12 ;  the  factor  3,  twice  in 
18  ;  and  the  factor  5,  once  in  30.  The  product  of  2  X  2  X  3 
X  3  X  5  is  the  least  common  multiple  required. 

What  is  the  least  common  multiple  of 

2.  12,  15,  and  20?  7.  24,72,18,48? 

3.  21,  24,  and  42?  8.  15,  24,  18,  32? 

4.  32,  48,  and  80?  9.  75,150,300? 

5.  27,  54,  and  108?  10.  125,  250,  500? 

6.  24,  80,  and  120?  11.  $48,  $7  2,  $144? 


PROPERTIES  OF  NUMBERS.  85 

Art.  53.  A  Multiple  of  a  number  is  any  number 
which  it  will  exactly  divide. 

Note. — Every  number  is  an  exact  divisor  of  its  product  by  an 
integer. 

A  Coinmon  Multiple  of  two  or  more  numbers 
is  any  number  which  each  of  them  will  exactly 
divide. 

The  Least  Common  Multiple  of  two  or  more 
numbers  is  the  least  number  which  each  of  them 
will   exactly  divide. 

Art.  54.  EuLE.— To  find  the  least  common  multiple 
of  two  or  more  numbers,  Resolve  the  numbers  into 
their  prime  factors,  and  then  select  all  the  different  fac- 
tors^ taking  each  the  highest  number  of  times  it  is  found 
in  any  number.  Multiply  the  factors  thus  selected ;  their 
product  will  be  the  least  common  multiple. 


Questions  for  Eeview. 

What  is  meant  by  the  divisor  of  a  number  ?  When  is  a 
divisor  a  common  divisor  ?  Define  a  common  divisor.  What 
is  the  greatest  common  divisor  of  two  or  more  numbers? 
How  is  it  found  ? 

By  what  may  every  number  be  divided  ?  What  is  a  prime 
number  ?  A  composite  number  ?  What  is  an  even  number  ? 
An  odd  number? 

What  is  meant  by  the  factor  of  a  number  ?  A  prime  fac- 
tor? A  composite  factor?  How  may  a  composite  number 
be  resolved  into  prime  factors? 

What  is  a  multiple  of  a  number?  When  is  a  multiple  a 
common  multiple?  Define  a  common  multiple.  What  is 
the  least  common  multiple  of  two  or  more  numbers?  Give 
the  rule  for  finding  the  least  common  multiple.  What  is 
the  differeuce  between  a  divisor  and  a  multiple^ 


SECTIOIS^  VII. 
I'HA  CTIOJTS. 


LESSON    I. 
The  Idea  of  a  JRractloji  developed. 

1.  If  a  melon  be  cut  into  two  equal  pieces,  what 
part  of  the  melon  will  one  piece  be? 

2.  How  many  halves  in  a  melon?  How  many 
halves   in   any  thing? 

3.  If  a  melon  be  cut  into  four  equal  pieces,  what 
part  of  the  melon  will  one  piece  be?  Two  pieces? 
Three   pieces? 

4.  How  many  fourths  in  an  apple?  How  many 
fourths   in   any   thing? 

5.  Which  is  the  greater,  one  half  or  one  fourth 
of  an  apple?     How  many  fourths  equal  one  half? 

6.  If  a  cake  be  cut  into  three  equal  pieces,  what 
part  of  the  cake  will  one  piece  be? 

7.  How  many  thirds  in  a  cake?  How  many  thirds 
in  an}^  thing? 

8.  If  a  cake  be  cut  into  six  equal  pieces,  what  part 

(86) 


P^RACTIONS.  87 

of  the  cake  will  one  piece  be?     Two   pieces?     Three 
pieces?     Four  pieces?     Five  pieces? 

9.  How  many  sixths  in  any  thing? 

10.  Which  is  the  greater,  one  third  or  one  sixth 
of  a  cake?     How  many  sixths  equal  one  third? 

11.  A  single  thing  is  a  unit.  How  many  halves 
in  a  unit?  How  many  thirds?  How  many  fourths? 
How  many  sixths? 

12.  What  is  meant  by  one  third? 

Ans.  One  third  is  one  of  the  three  equal  parts  of  a 
unit. 

13.  What  is  meant  b}^  two  thirds?  One  fourth? 
Three  fourths?     One  sixth?     Three  sixths? 

14.  Which  is  the  greater,  two  thirds  or  a  unit? 
Five  thirds  or  a  unit?  Three  fourths  or  a  unit? 
Four  fourtlis   or   a   unit? 

Art.  55.  Such  parts  of  a  unit  as  two  thirds,  three 
fourths,  five  sixths,  etc.,  are  called  Fractions. 

A  fraction  may  be  expressed  by  two  numbers,  one 
written  under  the  other,  with  a  horizontal  line  be- 
tween them;   as,  |,  f,  f. 

The  number  below  the  line  denotes  the  number 
of  equal  parts  into  which  the  unit  is  divided.  It 
is  called  the  Denominator. 

The  number  above  the  line  denotes  the  number 
of  equal  parts  taken.     It  is  called  the  Numerator, 

Eead  the  following  fractions.  How  is  the  unit  di- 
vided, and  how  many  parts  are  taken  in  each  case? 


(15) 

16) 

(17) 

(18) 

(19) 

(20) 

f 

f 

A 

A 

A 

H 

1 

A 

t\ 

H 

\% 

M 

1 

% 

5 

TT 

U 

M 

M 

88 


INTERMEDIATE  -  ARITHMETIC. 


Write  the  following  fractions  in  figures : 

(21)  (22)  (23) 

Two  fifths;  Seven  twelfths;  Twenty-four  fortieths; 

Seven  ninths;       Ten  thirteenths;  Thirty-five  fiftieths ; 

Nine  tenths  ;        Forty  fiftieths ;  Twenty -two  twelfths ; 

Ten  ninths.  Twenty  seventeenths.  Forty  fifty-fifths. 


DEHNITIONS. 

Art.  56.  A  Fraction  is  one  or  more  of  the  equal 
parts  of  a  unit. 

Art.  57.  A  fraction  is  expressed  by  two  numbers, 
called  the  Numerator  and  the  Denominator. 

The  Uenoininator  of  a  fraction  denotes  the  num- 
ber of  equal  parts   into  which   the  unit  is  divided. 

The  Nmnerator  of  a  fraction  denotes  the  number 
of  equal  parts  taken. 

The  numerator  and  denominator  are  called  the 
Terms  of  a  fraction. 


LESSON    II. 
Integers  mid  Mixed  JVumbers  reduced  to  I'r  act  ions. 

1.  How  many  thirds  in  an 
apple?  How  many  thirds  in 
2  apples? 

2.  How  many  fourths  in  3 
pears  ? 

Solution. — In  1  pear  there 
are    4    fourths,   and   in    3    pears 

there  are  three  times  4  fourths,  which  is  12  fourths.     There 
are  12  fourths  in  3  pears? 

3.  How  many  sixths  in  3  oranges?     In  5   oranges? 
6  oranges?     8  oranges? 


FRACTIONS. 


89 


4.  How  many  fifths  in  3?     5?     8?     10? 

5.  How  many  eighths  in  4?     6?     8?     10? 

6.  How  many  halves 
in  2  and  1  half  oranges? 

Solution.  —  In  2 
oranges  there  are  twice 
2  halves,  which  is  4 
halves,  and  4  halves 
and  1  half  are  5  halves. 
There  are  5  halves  in  2 
and  1  half  oranges. 


7.  How  many  fourths  in  2  and  3  fourths?      * 

8.  How  many  thirds    in  5  and  2  thirds?     8   and    1 
third?     7  and  2  thirds? 

9.  How  many  sixths   in  5  and   2  sixths?     8  and  3 
sixths?     12  and  5  sixths? 

10.  How  many  tenths   in    6   and  3  tenths?     7    and 
5  tenths?     8  and  7  tenths? 

11.  Eead6|;  33^;  45i ;  25|1;  50,^;  66^%. 


12.  How  many  fifths  in  6f  ?     8|? 


12|? 


WRITTEN  EXERCISES. 

13.  Eeduce  157  to  ninths.     157-Z-  to  ninths. 


157 
9 


1571 
9 


PROCESS : 

1413 
9 

,  Am. 

PROCESS : 

1413 

7 

1420 

,  Ans, 


14.  Eeduce  96  to  eighths.     96|  to  eighths. 


15.  Eeduce  35ii^  to  twelfths. 


46|-  to  ninths. 


90 


INTERMEDIATE  ARITHMETIC. 


16.  Reduce  73^^  to  elevenths.     63^  to  sevenths. 

17.  Eeduce  53|-J  to  a  fraction. 
Suggestion. — Reduce  the  mixed  number  to  twentieths. 

18.  Eeduce  33y^  to  a  fraction. 
Reduce  to  a  fraction: 


19.   85^\ 

22. 

236| 

25. 

48H 

20.   361^ 

23. 

49,^^ 

26. 

6VoV 

21.   4Sji 

2-1. 

^h'^ 

27. 

^hh 

To  Teachers. — See  Manual  of  Arithmetic  for  additional 
problems  in  this  and  the  following  lessons  in  Fractions. 

Art.  58.  A  Mixed  Number  is  an  integer  and  a 
fraction   united;    as,    51,    16|,   33^. 

Art,  59.  Rules. — 1.  To  reduce  an  integer  to  a  frac- 
tion, Multiply  the  integer  by  the  given  denominator,  and 
write  the  denominator  under  the  product, 

2.  To  reduce  a  mixed  number  to  a  fraction,  Multiply 
the  integer  by  the  denominator  of  the  fraction,  to  the 
product  add  the  numerator,  and  write  the  denominator 
under  the  result. 


LESSON    III. 
I^ractions  reduced  to  Integers  or  Mixed  JVumbers. 

1.  How  many  pears 
in  6  half-pears?  In  7 
half-pears  ? 

2.  How  many  days 
in  10  half-days?  In 
11   half-days? 

Solution. — In  11  half-days  there  are  as  many  days  as  2 
half-days  are  contained  times  in  11  half-dayS;  which  is  5 J 
times.     There  are  5J  days  in  11  half-days. 


FRACTIONS.  91 

3.  How  many  pints  in   14  half-pints?     In   17   half- 
pints?     In   21   half-pints? 

4.  How  many  yards   in    18   thirds   of  a  yard?     In 
19  thirds  of  a  yard?     In   22  thirds  of  a  yard? 

5.  How  many   weeks    in    28   sevenths   of  a  w^eek? 
30  sevenths  of  a  week? 

6.  A  mason  was  17  half-days  in  building   a   wall : 
how  many  days  did  he  work? 

7.  A  boy  earned  25  fourths  of  a  dollar  by  selling 
papers:    how  many  dollars  did  he  earn? 

8.  A  man  walked  25  eighths  of  a  mile  in  an  hour: 
how  many  miles  did  he  walk? 

9.  How  many  ones  in  -%5-?     ^-?     \^-?    -«/? 

10.  How  many  ones  in  f^?    "ff?    i^?    |3? 


WKITTEN   EXERCISES. 

11.  Eeduce  ^^^-  to  a  mixed  number. 

Process  :  W-  =  ^77  -:~  15  ^^  iif|,  Am. 

12.  Eeduce  ^^-  to  a  mixed  number. 
Eeduce  to  an  integer  or  mixed  number: 


13. 

n- 

17. 

M 

21. 

-w 

14. 

w 

18. 

-w 

22. 

w 

15. 

w 

19. 

W 

23. 

w 

16. 

¥/ 

20. 

-w 

24. 

w 

25.  li^ 

26.  V/ 

27.  -V_o 

28.  ^^ 


Art.  60.    A    Proper    Fraction    is    one    whose    nu- 
merator is  less  than   its  denominator;    as,   |,  |^,   |-. 

An   Improper  Fraction    is    one  whose  numerator 


92 


INTERMEDIATE   ARITHMETIC. 


is    equal   to    or   greater   than    the    denominator;    as, 

h  h  f- 

The  value  of  a  proper  fraction  is  less  than  one ;  and  the  value 
of  an  improper  fraction  is  equal  to  or  greater  than  one. 

Art.  61.  EuLE. — To  reduce  an  improper  fraction  to 
an  integer  or  mixed  number,  Divide  the  numerator  of 
the  fraction  by  the  denominator. 


LESSON   IV. 


JFh^actions  reduced  to  Zojper    Terms. 

1.  How  many  half-inches  in 
2  fourths  of  an  inch?  In  4 
fourths   of   an   inch? 

2.  How  many  thirds  of  an 
inch  in  2  sixths?  In  4  sixths? 
In  6  sixths? 

3.  How  many  fourths  in  6 
eighths? 

Solution. — In  2  eighths  there  is  1  fourth,  and  in  6  eighths 
there  are  as  many  fourths  as  2  eighths  are  contained  times  in 
6  eighths,  which  is  3.     There  are  3  fourths  in  6  eighths. 

4.  How  many  fifths  in  2  tenths?  In  4  tenths? 
6   tenths?     8   tenths?     12   tenths? 

5.  How  many  fifths  in  6  fifteenths?  9  fifteenths? 
12   fifteenths?     18   fifteenths? 

6.  How  many  sevenths  in  ^?     -i-l?     ^? 

7.  How  many  eighths  in  ^?     |f  ?     -|4? 

8.  Eeduce   ^,   ^,   and   ^   each   to   fourths. 

9.  Eeduce   If,   f|,   and   -^   each   to   sevenths. 
10.  Eeduce   -^-l,   -||-,   and   ^   each   to   eighths. 


FRACTIONS.  93 

,    and   II   each    to   sixths. 
12.  Keduce    |f ,   ^,   and   |-|   each   to   tenths. 


11.  Eeduce   |-|,   f^,   and   ||   each   to   sixths. 


WRITTEN   EXERCISES. 

13.  Reduce  ff  to  its  lowest  terms. 

PROCESS.  Reduce  f f  to  |J  by  dividing 

63^3       21-r-7       3      .  both  terms  by  3;  next  reduce 

34_^3~~28^  'T"  4 '         *        f  i  ^^  f  ^y  dividing  both  terms 

bv  7 ;  J  can  not  be  reduced  to 
6  3-^21       3  * 

Qj..     ^ — _  lower  terms,  and,  hence,  is  in 

8  4  -f-  2 1      4  its  lowest  terms.     Or,  reduce  f  j 

to  f  by  dividing  both  terms 
by  21,  the  greatest  number  which  will  exactly  divide  each 
term. 

Note. — The  teaclier  should  show  that  the  value  of  a  fraction  is 
not  ciianged  by  dividing  both  of  its  terms  by  the  same  number. 

Reduce  to  lowest  terms: 


14. 

n 

18. 

T% 

22. 

m 

26. 

m 

15. 

m 

19. 

2oG 

23. 

t'A 

27. 

ill 

16. 

96 
144 

20. 

tV% 

24. 

m 

28. 

m 

17. 

■rh 

21. 

A\ 

25. 

m 

29. 

m 

Art.  62.  When  a  fraction  is  reduced  to  an  equiv- 
alent fraction  with  smaller  terms,  it  is  reduced  to 
lower  terms. 

A  fraction  is  in  its  Lowest  Terms  when  no  integer 
except  1  will  exactly  divide  both  numerator  and  de- 
nominator. 

Art.  63.  Principle. —  The  division  of  both  terms  of  a 
fraction  by  the   same  number  does  not  change  its  value. 


94 


INTERMEDIATE  ARITHMETIC. 


Art.  64.  EuLE. — To  reduce  a  fraction  to  its  lowest 
terms,  Divide  both  tet^ms  of  the  fraction  by  any  common 
divisor;  then  divide  both  terms  of  the  residting  fraction 
by  any  common  divisor ;  and  so  on,  until  the  terms  of 
the  resulting  fraction  have  no  common  divisor  except  1. 

Or :  Divide  both  terms  of  the  fraction  by  their  great- 
est common  divisor. 


LESSON    V. 

Israel  Ions  reduced  to  Illghe?*  2*ertns» 

1.  How  many- 
fourths  of  an  or- 
ange in  1  half? 
In    2    halves? 

2.  ITow  many 
eighths  of  an  or- 
ange  in   1  fourth  ?     In   3   fourths  ? 

Solution. —  In  1  fourth  there  are  2  eighths,  and  in  8 
fourths  there  are  3  times  2  eighths,  which  is  6  eighths. 
There  are  6  eighths  in   3   fourths. 


3.  How    many    ninths    in    1    third?     In    2    thirds? 
3  thirds?    4  thirds? 

4.  How  many  tenths  in  |?     f?     |? 

5.  How  many  twelfths  in  f  ?     f  ?    |? 

6.  Change  \  and  ^  each  to  twelfths. 

7.  Change  f ,  |,  and  |-  each  to  eighteenths. 

8.  Change  -|,  \,  and  ii  each  to  twenty-fourths. 

9.  Change  |-,  -j7^,  and  -^  each  to  thirtieths. 

10.  Change  -f-,  -J-i,  and  \  each  to  twenty-eighths. 


FRACTIONS.  95 

WRITTEN  EXERCISES. 

11.  Change  H  to  seventieths. 

PROCESS.  One  thirty-fifth   is  as  many  seven- 

70^35  =  2.  tieths  as  35  is  contained  times  in  70, 

17V9      ^4  which  is  2  times,  and   17  thirty-fifths 

= — J  ^^^'  are  17  times  2  seventieths,  which  is  34 

^  seventieths.     This  is  the  same  as  mul- 
tiplying  both  terms  by  the  quotient 
of  70  divided  by  35. 

12.  Change  if  to  ninety-sixths. 

13.  Change  ^]-  and  ||  each  to  eighty-fourths. 

14.  Change  -^y  ^,  and  |-|  each  to  seventy-seconds. 

15.  Eeduce    f ,    |^,    and   \^   to    equivalent   fractions 
having  a  common   denominator. 

PROCESS.  ^, 

Change  the  fraction  to  twenty-fourths^ 


Heduce  to  a  common  denominator: 

IC.  11^         19.  f    I    t!^         22.    I    VV  A  tt 
17.  I     t     t  20.  I    f    Jj         23.    I     H  \l  If 

^^'    Z      lU    To  ^^-    H     T2-      2  4  ^^'    TXT      20      25^     FO 

Art.  65.  When  a  fraction  is  changed  to  an  equiva- 
lent fraction  with  greater  terms,  it  is  reduced  to 
Higher  Terms. 

Several  fractions  having  the  same  denominator,  are 
said  to  have  a  Common  Denominator. 

Art.  66.  Principle. —  TTie  rmdtiplication  of  both  terms 
of  a  fraction  by  the  same  number  does  not  change  its 
value. 


96  INTERMEDIATE   ARITHMETIC. 

Art.  67.  EuLES. — 1.  To  reduce  a  fraction  to  higher 
terms,  Divide  the  given  denominator  by  the  denominator 
of  the  fraction,  and  multiply  both  terms  by  the  quotient. 

2.  To  reduce  fractions  to  a  common  denominator, 
Divide  the  least  common  multiple  of  the  denominators 
by  the  denominator  of  each  fraction,  and  multiply  both 
terms  by  the  quotient, 

LESSON    VI. 
Addllion  of  JFractlo7is, 

1.  A  boy  gave  1  fourth  of  a  pine-apple  to  his 
brother,  1  fourth  to  his  sister,  and  1  fourth  to  a 
playmate:    what  part  of  it  did   he  give  away? 

How  many  fourths  are  ^  -f  ^  +  |-? 

2.  A  grocer  sold  1  eighth  of  a  cheese  to  one  cus- 
tomer, 2  eighths  to  another,  and  3  eighths  to  another: 
what  part  of  it  did  he  sell? 

How  much   is  I  +  I  +  I? 

3.  How  many  sixths  in  \,  |-,  and  f  ?  f ,  f ,  and  A? 
I-,  f,  and  f  ? 

4.  A  boy  gave  1  half  of  his  money  for  a  knife, 
and  1  third  of  it  for  a  ball:  what  part  of  his  money 
did  he  spend? 

Suggestion.— Change  \  and  \  to  sixths. 

5.  How  many  tenths  in  ^  and  |?    \  and  -j^? 

6.  How  many  twelfths  in  \  and  i?     \  and  |? 

7.  How  many  eighths  in  J  and  i^?     |  ynd  |? 

8.  How  many  fifteenths  in  \  and  i?  |  and  |? 
1-  and  f  ?    I  and  f  ? 


FRACTIONS.  97 

9.  How  many  twentieths   in  i  and  |?     |   and   |? 
f  and  I?     I  and  f ? 

10.  How  many  twenty -fourths  in  i,  ^,   \,  and   J? 
In  i     2.     3.     and  4 '<* 

WRITTEN    EXERCISES. 

11.  What  is  the  sum  of  ^3,  y^^,  ^,  -j^? 

PROCESS  :  A  +  A  +  A  +  A  ^  T  J  ^  2^ ,  ^^«. 

12.  What  is  the  sum  of  |f,  |f ,  -5^,  and  -if? 

13.  What  is  the  sum  of  |^,  if,  ff^  and  f|? 

14.  What  is  the  sum  of  f ,  |^,  and  j^? 

PROCESS.  Change  the  fractions  to  twenty- 

5  _|_    7  _|_   5   -_  fourths ;    add    the   numerators    of 

20   1    21    I    10 51  the  new  fractions;   and  reduce  the 

resulting    improper    traction   to    a 
ii'=^Ti  =  ^y  ^^^'  mixed  number. 

15.  What  is  the  sum  of  f ,  |,  and  ^? 

16.  Add    I,    I,    and    I . 

17.  Add    i,  ^,  and  i^. 

18.  Add    I,  ^,  and  ||. 

19.  Add    i,    f,    andij. 

20.  Add    I,  ^,    f,    and   if. 

21.  Add    f ,    I,    j\,  and  i|. 

22.  Add    f,  ^,-,  1^,  and   |i. 

23.  Add^,  H,  /^,  and   ||. 

24.  What  is  the  sum  of  16|,  18f ,  and  37|? 


First  add  the  fractions  and   then   the   in- 
tegers.     |  =  A,    i=--ft,   i^A-     A  +  « 


PROCESS. 

16f       A 

371        ?  4-  A  =  ft  =  IH  •     Write  the  ^  under  the 

- — '-  fractions  and  add  the  1  with  the  intecrers. 

I.  A.— 7. 


98  INTERMEDIATE  ARITHMETIC. 

25.  Add  45^-,  67|,  and  62f . 

26.  Add  37|,  18f ,  33|,  and  25-^, 

27.  Add  30^,  66|,  84f ,  and  133i. 

28.  Add  75|,  108,  160f ,  and  207. 

Art.  68.  EuLES.  —  1.  To  add  fractions,  Reduce  the 
fractions  to  a  common  denominator^  add,  the  numerators 
of  the  new  fractions^  and  under  the  sum  write  the  com- 
mon denominator. 

2.  To  add  mixed  numbers,  Add  the  fractions  and 
the  integers  separately^  and  combine  the  results. 


LESSON    VII. 

Subtraction  of  JFVactions, 

1.  Albert  had  2  thirds  of  an  orange,  and  he  gave 
1  third  to  his  sister:    how  many  thirds  had  he  left? 

How  much  is  |  less  i?     |  less  |? 

2.  Charles  bought  3  fourths  of  a  pound  of  raisins, 
and  then  gave  1  fourth  of  a  pound  to  his  playmate: 
what  part  of  a  pound  had  he  left? 

How  much  is  f  less  i?     f  less  |? 

3".  A  farmer  bought  |  of  a  bushel  of  flax-seed,  and 
sold  1^  of  a  bushel  to  a  neighbor:  what  part  of  a 
bushel  had  he  left? 


SuoGESTioN.— Change  \  and  \  to  sixths. 

4.  How  much  is  f  less  i? 

f  less  1? 

5.  How  much  is  \  less  |? 

\  less  1? 

6.  How  much  is  -^  less  \'l 

-h  l^ss  1? 

7.  How  much  is  -^  less  f  ? 

A  less  4? 

8.  How  much  is  if  less  f? 

H  less  f ? 

9.  How  much  is  \  less  -J-? 

A  less  3^? 

FRACTIONS.  99 


WRITTEN  EXERCISES. 

10.  From  ii-  take  ^^. 

PROCESS :  J^  —  ^  —  ^4^  =:  1 ,  Arts. 

11.  From  If  take  \\ . 

12.  From  ^\  take  ^\\. 

13.  From  i|  take  |. 

PROCESS  :   1 1  —  f  ^  1 2  _  1 5  ^  ^7^  ^  ^^5^ 

14.  Take  f  from  y7_;   |  from  f . 

15.  Take  -^-^  from  ^;   -J-^  from  f|. 

16.  Subtract  if  from  ff ;    |f  from  |f . 

17.  Subtract  y^^  from  -j^;   -j^  from  |-. 

18.  Subtract  y\  from  ^\ ;   ^  from  J-|. 

19.  Subtract  ^  from  -J-^]   \\  from  ii. 

20.  From  33^  take  18f . 

First  subtract  the  fractions  and  then  the 

PROCESS.  integers.     Since  x%  is  greater  than  ^^,  add 

33 J        ^         ^  to  j%,  making   ^f ,  and  then  take  the  ^ 

18|        f^         from  ^,  writing/^  under  the  fractions,  and 

14  7     j^g         adding  1  to  the  8  units   before   subtracting 

the  integers. 

21.  Take  30^  from  66|;   45f  from  66|. 

22.  Take  112|  from  145^;    90^  from  108|-. 

23.  Subtract  250f  from  300;    105|  from  261i. 

24.  Subtract  13o|  from  241f ;    166|  from  233|. 

Art.  69.  EuLES.  —  1.  To  subtract  fractions,  Beduce 
the  fractions  to  a  common  denominator^  subtract  the 
numerator  of  the  subtrahend  from  the  numerator  of  the 
minuend^  and  under  the  difference  write  the  common 
denominator. 

2.  To  subtract  mixed  numbers,  First  subtract  the 
fractions  and   then   the   integers. 


100  INTERMEDIATE  ARITHMETIC. 


LESSON    VIII. 

Problems  involving  the  Addition  aiid  Subtractio7i 
of  I^ractions, 

1.  A  boy  spent  \  of  bis  money  for  a  sled,  and  | 
of  it  for  a  pair  of  skates:    wbat  part  had  be  left? 

2.  John  bought  a  knife  for  f  of  a  dollar  and  a 
ball  for  1^  of  a  dollar,  and  then  sold  both  of  them 
for  |-  of  a  dollar :  what  part  of  a  dollar  did  he  gain  ? 

3.  Jane  having  |^  of  a  quart  of  plums,  gave  -|^  of  a 
quart  to  her  brother  and  |^  of  a  quart  to  her  sister: 
how  much  had  she  left? 

4.  A  farmer  bought  11  bushels  of  clover-seed,  and 
then  sold  |-  of  a  bushel  to  one  neighbor  and  f  of  a 
bushel  to  another:   what  part  of  a  bushel  had  he  left? 

5.  A  student  spends  \  of  his  time  in  study,  J^  of 
it  in  labor,  and  \  of  it  in  sleep:  what  part  has  he  left? 

6.  One  sixth  of  a  pole  is  in  the  ground,  |-  of  it  in 
water,  and  the  rest  in  the  air:  what  part  is  in  the  air? 

7.  .A  man  bequeathed  \  of  his  estate  to  his  wife, 
\  of  it  to  each  of  his  two  sons,  and  the  rest  to  liis 
daughter:    Avhat  part  did  the  daughter  receive? 

8.  A  man  did  \  of  a  piece  of  work  the  first  day, 
\  of  it  the  second  day,  and  then  completed  it  the 
third:    what   part   did    he   do   on    the  third    day? 

WRITTEisr   EXERCISES. 

9.  From  the  sum  of  |,  f ,  and  |  take  i^. 

10.  From  the  sum  of  |  and  \  take  their  difference. 

11.  A  man  owning  -y-  of  a  vessel,  sold  \  and  \ 
of  the  vessel:    what  part  had  he  left? 

12.  A  farmer  bought  240-|  acres  of  land,  and  sold 
90|-  acres  and  75^  acres :    how  many  had  he  left? 


FRACTI01S13.  101 

13.  From  a  piece  of  broadcloth  Containing  20| 
yards,  a  merchant  sold  5|-  yards,  4^  yards,  and  8^ 
yards:    how  many  yards  were  left? 

14.  A  man  earned  $56|-  one  month  and  $70|  the 
next,  and  then  gave  $85^  for  a  horse :  how  much 
money  had  he  left? 

15.  From  47f  +  33^  take  their  difference. 

16.  A  pedestrian  walked  -f-^  of  his  journey  the 
first  day,  ^  of  it  the  next  day,  and  completed  it 
the  third  day:  what  part  of  the  journey  did  he 
travel  the  third  day? 


LESSON    IX. 

J^ractio7is  JVfult /plied  by  Integers. 

1.  What  j^art  of  a  cake  is 
twice  2  eighths  of  it?  3  times 
2  eighths  of  it? 

2.  A  father  gave  3  fourths 
of  an  orange  to  each  of  4 
children :  how  many  fourths 
did   they  all   receive? 

3.  How  much  is  4  times  |?     6   times  f  ? 

4.  If  a   boy    earn    2   thirds   of  a  dollar    in    a   day, 
how  much  will   he  earn  in  3  days? 

5.  How  much  is  3  times  |?       5  times  |? 

6.  IIow  much  is  6  times  f  ?       9  times  f  ? 

7.  How  much  is  7  times  |?       8  times  |? 

8.  How  much  is  5  times  6|?     7  times  8-i-? 

Suggestion. — Multiply  the  integer  and  the  fraction  separately, 
and  add  the  products. 

9.  How  much  is  3  times  6|?     8  times  7|? 

10.  How  much  is  6  times  4f  ?     9  times  Sf? 

11.  How  much  is  8  times  6|?     8  times  7f  ? 


102 


INTERMEDIATE  ARITHMETIC. 


Multiply : 


AATRITTEN  EXERCISES. 


12.  T-V  by  8. 

15. 

li  by  25. 

•18. 

if  by  16, 

13.  ^,  by  12. 

16. 

ii  by  16. 

-19. 

16|  by  4. 

14.  If  by  24. 

17. 

«  by  33. 

•20. 

18|  by  12. 

Art.  70.  Principle. — A  fraction  is  multiplied  by  mul- 
tiply ing  its  numerator  or  dividing  its  denominator. 

Art.  71.  EuLES. — 1.  To  multiply  a  fraction  by  an 
integer,  Multiply  the  numerator  or  divide  the  denom- 
inator. 

2.  To  multiply  a  mixed  number  by  an  integer,  Mul- 
tiply the  fraction  and  the  integer  separately ^  and  add 
the  products. 

LESSON    X. 


JRr actional  !Pa?^is  of  Integers. 

1.  If  6  pears   be  , divided   equally  between  2  boys, 
what  part  of  the  whole  will  each  receive? 

What  is  \  of  6 
pears?  \  of  10 
pears  ? 

2.  A  father  di- 
vided 5  melons 
equally  between 
2  children  :  what 
part  of  the  whole 
did  each  receive? 

What  is  I  of  5 
melons? 


Suggestion.— Take  1  half  of  4  melons  and  then  1  half  of  1  melon. 


FRACTIONS.  103 

3.  Charles    divided    12    plums    equally    between    3 
boys:   what  part  of  the  whole  did  each  receive? 


4. 

What  is  ^  of  9? 

^  of  12? 

i  of  16? 

5. 

What  is  ^  of  20? 

\  of  28? 

1^  of  30? 

6. 

What  is  1  of  25? 

1  of  26? 

I  of  37? 

7. 

What  is  J-  of  24? 

f  of  24? 

Solution. — J  of  24  is  4,  and  f  of  24  is  5  times  4,  which 
is  20.    f  of  24  is  20. 

8.  What  is  -^  of  40?    ^  of  40?    ^  of  40? 

9.  What  is  f  of  63?       f    of  64?    f  of  65? 

10.  What  is  I  of  45?       f   of  37?    |  of  58? 

11.  What  is  I  of  33?       I    of  58?    ^  of  50? 


WRITTEN  EXERCISES. 

12.  What  is  I  of  659^ 

8)659  659 


PROCESS  :  o  Z  f 


824  Or:  ^ 


3  8)1977 


247i,Ans.  247^ 

13.  What  is  |  of  191?    |  of  367? 

14.  What  is  f  of  508?    ^  of  243? 

15.  What  is  f  of  466?    ^  of  4648? 

16.  What  is  ^  of  906?    f|  of  6070? 

Integers  tnuWpUed  by  I^ractfons, 

17.  Multiply  256  by  f . 

f  is  3  times  J,  and  hence 
4)256  256       3  times  256  is  3  times  \  of 


64  ^      256.      Or,  f  is  4  of   3,   and 

3  ^^ •     4 )768      hence  f  times  256  is  J  of  3 

192,  Arts.  192       times  256. 


104  INTERMEDIATE  ARITHMETIC. 

18.  48  by  yV  22.  163  by  -^ ,  "  26.  248  by  if . 

19.  65  by    I-.  23.  300  by  if.  '27.  406  by  ^. 

20.  59  by    f  .  ^  24.  257  by  fi .  28.  856  by  ^f  . 
-^21.  87  by    |.  ^25,  305  by  |^.  29.  794  by  |f . 

30.  Multiply  324  by  16f . 

324 

1^1  First  multiply  by  the 

1944  integer  and  then  by  the 

process:             3  24  fraction,    and    add    the 

2  16  products. 
5400,  Ans. 

/31.  48  by  16|.        34.  246  by  12|.        37.  108  by  56f. 

'  32.  72  by  18f .        35.  324  by  17i.        38.  524  by  72f . 

33.  96  by  23^ .        36.  406  by  S3i .        39.  684  by  66| . 

Art.  72.  EuLES. — 1.  To  find  the  fractional  part  of  an 
integer,  or  to  multiply  an  integer  by  a  fraction,  Divide 
the  integer  by  the  denominator  and  multiply  the  quotient 
by  the  numerator. 

Or:  Multiply  the  integer  by  the  numerator  and  divide 
the  product  by  the  denominator. 

2.  To  multiply  an  integer  by  a  mixed  number.  Mul- 
tiply by  the  integer  and  the  fraction  separately ,  and  add 
the  products, 

LESSON    XI. 

Coinpoii7id  I^racUons  reduced  to  Simple  J^7*acHo7is, 

1.  A  boy  having  i  of  a  melon,  gave  -^  of  it  to  a 
playmate:    what  part  did  the  playmate  receive? 
What  is  i  of  ^?     i  off? 


FRACTIONS.  105 

2.  If  each  third  of  a  pine -apple  bo  cut  into  2 
equal  pieces,  what  part  of  the  pine-apple  will  1 
piece   be? 

What  is  i  of  i? 


3.  What  is  i  of  i?    ^  of  ^?    ^  of  i? 

4.  What  is  i  of  i?     ^  of  i?     i-  of  i? 

5.  What  is  ^  of  i?    i  of  |?     -^  of  -i? 

6.  A  girl  having  f  of  an  orange,  divided  it  equally 
between  her  2  brothers :  what  part  of  the  orange 
did  each  receive? 

Suggestion. — Divide  each  fourth  into  2  equal  pieces,  and  then 
give  3  pieces  to  each. 


7.  What  is  ^  of  ^?    ^  off? 

8.  What  is  i  of -i?     i  of 


3.? 

8  • 


ioff? 


9.  What  is  I  of  I?     i  of  f  ?     I  of  I? 
Solution. — J  of  f  is  ^^,  and  j  of  f  is  3  times  ^%,  which 

lb  ^ry.       :f   OI    ^  —  ^^. 

10.  What  is  I  of  f  ?     I  of  I?     f  of  I? 

11.  What  is  f  of  -I?     I  of  f  ?     I  of  3^? 

12.  What  is  i  of  12^?     i  of  131? 

Solution.— J  of  13^  ==  ^  of  12  +  J  of  li  or  f .    J  of  12  is 
4,  and  J  of  f  is  f  or  | .     Hence,  J  of  13|  is  4^ . 

13.  What  is  I  of  18|?    i  of  21|?     |   of  31|? 

14.  What  is  |  of  22|?     i.of  42^?      i    of  46f  ? 

15.  What  is  ^  of  334?     I'of  644?     ^^^  of  62|? 


106  INTERMEDIATE  ARITHMETIC. 

AVKITTEN  EXERCISES. 

16.  Reduce  f  of  |  to  a  simple  fraction. 

2        3_2><^__6__2    ^^^ 
process:     gOt  5-3^5  ~15~5' 


2     .  3      2X$ 


0 


Note.  —  The  examples  should  be  solved  by  both  methods.    The 
teacher  should   explain  the  process  of  cancellation. 


Eeduce  to  a  simple  fraction : 


17. 

f  of  i. 

21. 

1    of  A. 

18. 

fof  f- 

22. 

f   of|f- 

19. 

f  of,^. 

23. 

Aofj^. 

20. 

|of  A- 

24. 

Aofif. 

25.  f  of  f  off. 

26.  I  of  f  of  2I-. 

27.  f  of  2|  of  I . 

28.  4  of  f  of  If . 


JRracti07is  niulilpUed  by  JFractlons* 

29.  Multiply  |  by  f. 

PROCESS.  Since  J  is  }  of  one,  f 

4X3       12       3         \Am^%  4  ^|  of  once  f,  or 

J  of  f ,  which  equals  — -^ — 

4X5 

^  4^3   4X3   3 

5X^"5'        ^^"^^'5>^4^.^r^  =  5' 

30.  I  by  |.     34.  if  by  3^.     38.  2i  by  2i. 

31.  f  by  ^.     35.  I^by  f4.     39.  3^  by  3^. 

32.  ,3^  by  |.     36.  I  by  ^-.     40.  61  by  2^. 

33.  I  by  3^.  37.  i^  by  V-.     41.  6^  by  12|. 


FRACTIONS.  107 

Art.  73.  A  Simple  Fraction  is  a  fraction  not  united 
with  an  integer  or  another  fraction ;    as,  f . 

A  Compound  Fraction  is  a  fraction  of  a  fraction ; 
as,  I  off;   |of  3i. 

Art.  74.  Rule. — To  reduce  a  compound  fraction  to 
a  simple  fraction,  or  to  multiply  one  fraction  by  an- 
other. Multiply  the  numerators  together,  and  also  the 
denominators. 

Note.— The  process  may  often  be  shortened  by  canceling  com-%k 
mon  factors  before  multiplying. 


LESSON    XII. 
J^rac lions  divided  by  l7itegers, 

1.  A  father  divided  J  of  a  melon  equally  between 
3  boys :    what  part  of  the  melon  did  each  receive  ? 

Solution. — If  3  boys  receive  f  of  a  melon,  each  will  re- 
ceive J  of  f ,  which  is  \, 

2.  A  woman  divided  f  of  a  pound  of  crackers 
equally  between  3  poor  children :  what  part  of  a 
pound  did  each  receive? 

3.  If  5  pounds  of  cheese  cost  f  of  a  dollar,  what 
will  1  pound  cost? 

4.  How  much  is  f  --  5?     f  -f-  5?     |  -f-  5? 

5.  If  6  men  can  build  |^  of  a  wall  in  a  day,  what 
part  of  the  wall  can  1  man  build? 

6.  How  much  is  |-  --  6  ?     |^  -^  5  ?     |  -'r-  9  ? 

7.  A  grocer  put  161  pounds  of  sugar  into  5  equal 
parcels:   how  much  sugar  was  put  into  each  parcel? 

8.  Divide  16^  by  5.  ^  12^  by  5.     181  by  5. 

9.  Divide  20|  by  3.     301  by  4.     31f  by  6. 


108 


INTERMEDIATE   ARITHMETIC. 


A^V^KITTEnsr  EXERCISES. 

10.  Divide  ^  by  3. 

PROCESS :  ^^~Z  =  \oi  ^^  =  /^ 

12  12X3      36 


11.  xVby    7. 

12.  6    by  12. 
13. 
14. 


6 

tV  V  10. 

T^by    6. 


15.  -If  by  7. 

16.  if  by  8. 

17.  If-  by  5, 
IS.fiby  3. 


19.  241  by  6. 

20.  29f  by  7. 

21.  46f  by  5. 

22.  66|  by  8. 


Art.  75.  Principle. — A  fraction  is  divided  by  dividing 
the  numerator  or  multiplying  the  denominator. 

Art.  76.  EuLES. — 1.  To  divide  a  fraction  by  an  in- 
teger, Divide  the  numerator  or  multiply  the  denominator. 

2.  To  divide  a  mixed  number  by  an  integer,  Divide 
the  integral  part  and  then  the  fraction. 


LESSON    XIII. 
I7itegers  divided  by  JFr actions. 


1.  How  many  times  is  \  of 
an  apple  contained  in  2  ap- 
ples? \  of  an  apple  in  2  ap- 
ples? 

2.  How  many  times  is  \  of 
a  yard  contained  in   3  yards? 


Solution. — 3  yards  =  12  fourths,  and  3  fourths    is    con- 
tained in  12  fourths  4  times. 


r 


FRACTIONS.  109 


3.  If   a   basket     hold     |^    of   a    bushel,    how    many 
baskets  will  hold  4  bushels? 

4.  How  many  times  is  |  contained  in  4?     |  in  4? 


f  in  4? 


5.  If  I  of  a  yard  of  silk  will  trim  a  hat,  how  many 
hats  will  6  j^ards  trim? 

G.  How  many  times  is  f  contained  in  3?  f  in  6? 
f  in  6?     f  in  3? 

7.  Divide  12  by  f ;     15  by  f ;     20  by  f . 

8.  Divide     8  by  A;     lObyf;     12  by  ^\. 

WRITTEN   EXERCISES. 

9.  Divide  14  by  f . 

PROCESS :   14  =  7^0-.     7^0  ^  3  ^  70  ^  3  =  23 J,  Ans, 

Note. — Since  14  is  reduced  to  fifths  by  multiplying  it  by  5, 
tlie  process  may  be  shortened  by  omitting  the  denominators,  thus: 
14^  f  =  14X5-f-3==23i. 

10.  16  by  |.  13.  60  by  f .  16.  30  by  2| . 

11.  20  by  I-.  14.  21  by  f.  17.  40  by  3^. 

12.  45  by  f .  15.  42  by  f .  18.  16  by  5i. 

Art.  77.  EuLE. — To  divide  an  integer  by  a  fraction, 

Multiply  the  integer  by  the  denominator  of  the  fraction, 
and  divide  the  product  by  the  numerator. 


LESSON     XIV. 
I^ractlons  divided  by  J^ractions, 

1.  If  1^  of  a   barrel   of  flour   will   supply  a  family 
1  month,  how  many  months  will  f  of  a  barrel  last  ? 

2.  How  many  times  is  \  contained  in  |-?    -J-  in  |? 


110  INTERMEDIATE    ARITHMETIC. 

3.  If  f  of  a  3'ard  of  cloth  will  make  a  vest,  how 
many  vests  will  |^  of  a  yard  make? 

4.  How  many  times  f  in  |^?     |^  in  |^? 

5.  If  a  pound  of  butter  cost  ^  of  a  dollar,  how 
many  pounds  can  be  bought  for  f  of  a  dollar? 

Suggestion.— Change  one  fourth  to  eighths. 

6.  How  many  times  ^  in  |^?     i  in  |^? 

7.  How  many  times  ^in|^?     iini?     |^ini? 

8.  How  many  times  iin|?     |^in|?     fin|? 

WKITTEN  EXEKCISES. 

1.  Divide  f  by  |. 

process:  }  =  a •    f  =  1%^  •    A  -^  A  =  f  =  IJ • 

5.  f  by  |.  8.  2|  by    |. 

6.  I  by  |.  9.  3^  by    f . 

7.  I  by  f .  10.     I  by  2|. 

Art.  78.  EuLE. — To  divide  a  fraction  by  a  fraction, 
Beduce  the  fractions  to  a  common  denominator^  and  di- 
vide the  numerator  of  the  dividend  by  the  numerator  of 
the  divisor. 

Note. — When  pupils  are  familiar  with  this  method,  they  may 
be  taught  to  divide  by  inverting  the  terms  of  tlie  divisor  and  mul- 
tiplying. This  method  is  fully  explained  in  the  author's  Com- 
plete Aeithmetic. 


LESSON    XV. 

JViimbers  J^ractlonal  'Parts  of  Other  JVumbers. 

1.  5    is   -i    of  what    number? 
SoLUTioi^'. — 5  is  I  of  3  times  5,  or  15. 


2. 

1  byf- 

3. 

1  by  i- 

4. 

f\  by  |. 

7. 

12 

is 

1 

of 

what 

8. 

25 

is 

1 

of 

what 

9, 

30 

is 

f 

of 

what 

10. 

33 

is 

f 

of 

what 

11. 

44 

is 

* 

of 

what 

FRACTIONS.  Ill 

2.  7    is    ^    of  what,  number? 

3.  12    is    j\   of  what    number? 

4.  12^    is    ^   of  what    number? 

5.  16|    is    |-    of   what    number? 

6.  10    is    I    of  what    number? 

Solution. — If  10  is  f  of  a  number,  -J  of  the  number  is  J 
of  10,  which  is  5 ;  if  5  is  J  of  a  number,  J  of  it  will  be  3 
times  5,  which  is  15. 

number? 
number? 
number? 
number? 
number? 

12.  A  man  spent  f  of  his  money  and  had  $21  left: 
how  much  money  had  he  at  first? 

13.  A  boy  gave  24  cents  for  a  slate,  which  was 
^  of  his  money:   how  much  money  did  he  have? 

14.  A  man  pays  $25  a  month  for  house-rent,  which 
is  -^-^  of  his  monthly  salary:   what  is  his  salary? 

15.  A  farmer  sold  a  cow  for  $45,  w^hich  was  \  more 
than  he  paid  for  her :   what  was  the  cost  of  the  cow  ? 

16.  A  man  sold  f  of  a  farm  for  $1500:  at  this  rate, 
what  was  the  value  of  the  farm? 


LESSON    XVI. 
Miscellaneous   'Problems, 

1.  Eeduce  18f  to  an  improper  fraction. 

2.  Eeduce  ^^-  to  a  mixed  number. 

3.  Eeduce  f ,  f ,  and  ^  to  a  common  denominator. 

4.  Add  i,  i,  I,  and  ^. 

5.  Add  28f ,  40 1-,  63|,  and  19^3^. 


112  INTERMEDIATE   ARITHMETIC. 

6.  From  f  take  f .     From  28|  take  16|. 

7.  Multiply  I  by  7;    13  by  |;    f  by  f. 

8.  Multiply  137|  by  15;   256  by  21|. 

9.  Divide  12  by  | ;   |  by  12;   f  by  f. 

10.  Divide  243|  by  11 ;    256  by  ^, 

11.  |  +  |  =  what?    f-l?     |x|?    i^i? 

12.  There  are  5280  feet  in  a  mile :  how  many  feet 
in  ^  of  a  mile  ? 

13.  A  vessel  is  worth  $6000,  and  the  cargo  is  worth 
I  as  much  as  the  vessel:  what  is  the  value  of  the 
cargo  ? 

14.  A  man  sold  |  of  his  farm  to  one  neighbor 
and  I  of  it  to  another:  what  part  of  the  farm  has 
he  left? 

15.  A  man  owning  -|  of  a  factory  sold  |  of  his 
share :  what  part  of  the  factory  did  he  sell  ?  What 
part  does  he  still  own? 

16.  There  are  161  feet  in  a  rt)d :  how  many  feet 
in  66  rods? 

17.  There  are  5^  yards  in  a  rod:  how  many  rods 
in  66  yards? 

18.  At  $6^  a  barrel,  how  many  barrels  of  flour  can 
be  bought  for  $150? 

19.  If  f  of  a  ship  is  worth  $12000,  what  is  the 
whole  ship  worth  ? 

20.  A  man  owning  ^  of  an  estate  sells  t  of  his 
tshare  for  $2400:  at  this  rate,  what  would  be  the 
value  of  the  estate  ? 

21.  A  farmer  had  2  fields  of  wheat;  the  first  yielded 
840  bushels,  which  was  ^  of  the  amount  yielded  by 
the  second  field :  how  many  bushels  did  the  second 
field  yield? 

22.  A  man  owning  |-  of  a  ship  sells  |-  of  his  share 
for  $4400 :  at  this  rate,  what  is  the  value  of  the  ship? 


FRACTIONS.  113 

28.  The    value    of  a   certain   shii^   is   $9760,  and    | 

of   the    value   of  the   ship   is   |^   of  the   value  of  the 
cargo :    what  is  the  value  of  the  cargo  ? 


Questions  for  Eeview. 

What  is  a  fraction  ?  What  does  the  denominator  denote  ? 
The  numerator?     What  are  the  terms  of  a  fraction? 

What  is  a  mixed  numher?  What  is  meant  by  18|?  Ans. 
18  -f-  J-  When  is  a  fraction  called  proper?  When  improper? 
When  is  the  value  of  an  improper  fraction  equal  to  1  ? 

How  is  an  integer  reduced  to  a  fraction  ?  How  is  a  mixed 
number  reduced  to  a  fraction  ?  What  kind  of  a  fraction  is 
the  result?  Give  examples.  How  is  an  improper  fraction 
reduced  to  a  whole  or  mixed  number  ? 

How  is  a  fraction  reduced  to  lower  terms?  On  what 
principle  does  the  process  depend?  How  may  a  fraction 
be   reduced   to  its  lowest  terms  by   one   division? 

How  are  fractions  having  a  common  denominator  added  or 
subtracted?  When  fractions  have  different  denominators, 
how  are  they  added  or  subtracted  ?  How  may  mixed  num- 
bers be  added?     How  may  they  be  subtracted? 

In  what  two  ways  may  a  fraction  be  multiplied  by  an 
integer?  How  may  an  integer  be  multiplied  by  a  fraction? 
Give  the  rule  for  multiplying  a  fraction  by  a  fraction. 
What  is  a  compound  fraction  ?  How  is  a  compound  frac- 
tion  reduced  to   a  simple   fraction  ? 

In  what  two  ways  may  a  fraction  be  divided  by  an  in- 
teger? How  may  an  integer  be  divided  by  a  fraction? 
How  may  a  fraction   be  divided   by  a  fraction  ? 

N.  B. — If  it  is  thought  best  to  teach  United  States  Money  before 
Decimal  Fractions^  introduce  Section  IX  at  this  point. 


1.  A.— 8, 


SECTION   VIII. 


LESSON    I. 
JVumeratJon  and  JVotation, 

1.  If  a  unit  be  divided  into  ten  equal  parts,  what 
is  one  part  called? 

2.  If  a  tenth  of  a  unit  be  divided  into  ten  equal 
parts,  what  is  one  part?     What  is  ^  of  y^^? 

3.  If  a  hundredth  of  a  unit  be  divided  into  ten 
equal  parts,  what  is  one  part?     What  is  J^  of  y^? 

4.  What  part  of  one  tenth  is  one  hundredth? 
What   part   of  one    hundredth    is   one    thousandth? 

5.  How  do  one  tenth  and  one  hundredth  compare 
with  each  other  in  value?     ^V  ^^^  rlo^ 

6.  How  do  one  hundredth  and  one  thousandth 
compare  with  each  other  in  value?     y|^  and  y (jV(r  ^ 

7.  How,  then,  do  the  fractional  units,  tenths,  hun- 
dredths, and  thousandths,  compare  in  value? 

Art.  79.  The  fractional  units,  tenths,  hundredths, 
and  thousandths  may  be  expressed  on  a  scale  of  tens, 
by  writing  the  tenths  in  the  first  order  at  the  right 
of  units,  the  hundredths  in  the  second  order,  and  the 
thousandths  in  the  third  order,  and  placing  a  period 
between  the  orders  of  units  and  tenths,  to  distinguish 
the  fractional  orders  from  the  integral  orders. 

Thus,   2.5  denotes  two  units  and   five  tenths;    4.06 
denotes   four  units   and    six   hundredths;    .05    denotes 
five  hundredths;  and  .004  denotes  four  thousandths. 
(114)        - 


DECIMAL   FRACTIONS.  115 

8.  How  many  tenths  in  .3?     In  .6?     In  .7? 

9.  How  many  hundredths  in  .02?     In  .04?     .07? 

10.  How  many  thousandths  in  .005?     In   .008? 

11.  How  many  tenths  and  hundredths  in  .24?    .06? 

12.  How  many  tenths,  hundredths,  and  thousandths 
in  .356?     In  .523?     In  .603?     In  .041? 

Art.  80.  Two  tenths  and  five  hundredths  (25)  de- 
note twenty-five  hundredths;  and  two  hundredths  and 
five  thousandths  (.025)  denote  twenty-five  thousandths. 

13.  How  many  hundredths  in  .34?    In  .45?    In  .06? 

14.  How  many  thousandths  in  .246?  In  .048?  In 
.605?     In  .007?     In  .403?     In  .075? 

Art.  81.  Such  fractional  numbers  as  .24  and  .208 
are  called  Decimal  Fractions,  or  Decimals,  and 
the  orders  of  which  they  arc  composed  are  called 
Decimal  Orders. 

The  first  decimal  order  is  tenths;  the  second,  hun- 
dredths;  and  the  third,  thousandths. 

Copy  and  read  the  following  decimals: 


(15) 

(16) 

(17) 

(18) 

(19) 

.12 

.014 

.324 

.53 

.004 

.3  4 

.06  3 

.406 

.6 

.803 

.0  6 

.008 

.065 

.009 

.550 

.50 

.030 

.704 

.057 

.400 

Art.  82.  When  fractions  denoting  tenths,  hun- 
dredths, thousandths,  etc.,  are  expressed  on  a  scale 
of  ten,  they  are  said  to  be  expressed  decimally.  The 
right-hand  figure  is  written  in  the  order  indicated 
by  the  name  of  the  decimal. 

20.  Express  decimally  25  hundredths. 

21.  Express  decimally  205  thousandths. 


116  INTERMEDIATE  ARITHMETIC. 

22.  Express  decimally  26^  thousandths. 

23.  Express  decimally  6J-  hundredths. 

24.  Express  decimally  forty-five  hundredths. 

25.  Four  hundred  and  twelve  thousandths. 

26.  Seven  hundred  and  eight  thousandths. 

27.  Sixty-five  thousandths. 

28.  Eight  units  and  seven  tenths. 

29.  Fifteen  units  and  thh^ty-six  hundredths. 

Art.  83.  The  fourth  decimal  order  is  called  ten- 
thousandths;  the  fifth,  hundred-thousandths;  and  the 
sixth,  millionths. 

Copy  and  read : 


(30) 

(31) 

(32) 

(33) 

.445 

.0304 

.3256 

.00267 

.0445 

.00304 

.4048 

.000267 

.706 

.475 

.03256 

.004324 

.0706 

.00475 

.04048 

.046  3  75 

34.  Express  decimally  3205  ten-thousandths. 

35.  Express  decimally  6008  hundred-thousandths. 

36.  Express  decimally  40532  millionths. 

37.  Two  hundred  and  seventeen  ten -thousandths. 

38.  Four  hundred  and  twenty -two  millionths. 

39.  Seven  hundred  and  twelve  ten-thousandtlis. 

40.  Fifteen  millionths. 

41.  Four  hundred  and  one  hundred-thousandths. 

Art.  84.  An  integer  and  a  decimal,  written  together 
as  one  number,  are  connected  by  and  when  expressed 
in  words.  Thus,  45.14  is  read  forty -five  units  and 
fourteen  hundredths. 

42.  Head    27.305.  44.  Eead    7.06005. 

43.  Eead    463.3028.  45.  Eead    4000.004. 


DECIMAL  FRACTIONS.  117 

46.  ExpresB  decimally  forty-five  units  and  fifty -two 
hundredths. 

47.  Forty  units  and  forty-five  thousandths. 

48.  Two  hundred  units  and  seventy-nine  millionths. 


DEFINITIONS,  PEINOIPLES,  AND  EULES. 

Art.  85.  A  Decimal  Fraction  is  a  fraction  whose 
denominator  is  ten  or  a  product  of  tens. 

The  decimal  denominators  are  10,  100,  1000,  10000,  etc. 

Art.  86.  Decimal  fractions  may  be  expressed  in 
three  ways: 

1.  By  words;  as,  three  tenths,  twelve  hundredths. 

2.  B}"  writing  the  denominator  under  the  numer- 
ator ;  as,  y%,  yV%. 

3.  By  omitting  the  denominator  and  writing  the 
numerator  decimally;  as,  .3  and  .12. 

Three  tenths,  -f^^  and  .3  express  the  same  decimal  fraction, 
but  the  term  decimal  is  usually  applied  to  a  decimal  fraction  when 
expressed  by  the  third  method.  Decimal  fractions  may  be  read  or 
dictated,  and  hence  may  be  expressed  iyi  words. 

Art.  87.  The  Decimal  Boiiit  is  a  period  placed 
at  the  left  of  the  order  of  tenths,  to  designate  the 
decimal  orders. 

Art.  88.  A  Mixed  Decimal  Number  is  an  integer 
and  a  decimal  written  together  as  one  number;  as, 
2.45.     It  is  also  called  a  Mixed  Kuiriber. 

The  orders  on  the  left  of  the  decimal  point  are 
integral^  and  those  on  the  right  are  decimal.  The 
decimal  orders  are  called  Decimal  Places. 


118-  INTERMEDIATE   ARITHMETIC. 

Art.  89.  The  following  table  gives  the  names  of  six 
inte«:ral  and  six  decimal  orders: 


n^ 

n3 

fl 

a 

^ 

OD 

^ 

s 

oc 

*fco 

13 

^ 

o 

-^ 

.g 

{» 

^ 

o 

ri:5 

S 

s 

j» 

^ 

G 

,£5 

O 

CO 

1 

02 

CO 

•1 

•5 

1 

O 
4^ 

CO 

.2 

K 

O 

i 

p 

s 

^ 

5 

o 

2 

w 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Iniegral  Orders.  Decimal  Orders. 

Art.  90.  Principles.— 1.  The  denominator  of  a  deci- 
mal fraction  is  1  ivith  as  many  ciphers  annexed  as  there 
are  decimal  'places    in    the  fraction. 

2.  The  successive  decimal  orders  decrease  in  value 
from  left  to  rights  and  increase  from  right  to  left  in 
the  same  manner  as  integral  orders. 

3.  The  name  of  a  decimal  is  the  same  as  the  name  of 
its   right-hand  order. 

Art.  91.  EuLES. — 1.  To  read  a  decimal,  Bead  it  as 
though  it  were  an  integer^  and  add  the  name  of  the  right- 
hand  order. 

Notes. — 1.  A  decimal  is  read  precisely  as  it  would  be  were  the 
denominator  expressed. 

2.  In  reading  a  mixed  decimal  number,  the  word  ''units"  may 
be  omitted  when  this  does  not  change  the  mixed  number  to  a 
pure  decimal. 

2.  To  write  a  decimal,  Write  it  as  an  integer^  and 
so  place   the  decimal  point  that  the   right-hand  figure 


DECIMAL  FRACTIONS.  119 

shall  stand  in  the  order  denoted  by  the  name  of  the 
decimal. 

Note.— When  tlie  number  does  not  fill  all  the  decimal  places, 
supply  the  deficiency  by  j^relixing  decimal  ciphers. 

W^RITTEN     EXERCISES. 

49.  Write  in  words  4045.03007. 

50.  Write  in  words   .040085. 

51.  Wiite  in  words  405.40071. 

52.  Write  in  words  35000.0094. 

53.  Express   decimally   five    thousand   and   sixty-six 
milliontbs. 

54.  Eight  hundred   and  forty-two  ten-thousandths. 

55.  Seventy-five   and  four  hundred  and  three  hun- 
dred -thousandths. 

To  Teachers. — See  Manual  of  Arithmetic  for  addi- 
tional problems  in  Decimal  Fractious. 


LESSON    II. 

^educfio7i  of  decimals. 

Case  I.  —  Decimals   Reduced   to   Higher  or  Lower 
Orders. 

1.  How  many  hundredths  in  1  tenth?     In  3  tenths? 
In  5  tenths?     In  .8? 

2.  How  many  thousandths  in   1    hundredth?     In  5 
hundredths?     In  .12?     In  .25? 

3.  How    many    tenths    in    10    hundredths?     In    40 
hundredths?     In  .50?     In  .60?     In  .80? 

4.  How  many  hundredths    in    10    thousandths?     In 
30  thousandths'?     In  .060?     In  .120?     In  .340? 


120  INTERMEDIATE  ARITHMETIC. 

WRITTEN   EXERCISES. 

5.  Eeduce  .325  to  hundred-thousandths. 

PROCESS.  According   to    Art.    66,    .32*5   or 

.3  2  5  =  .3  2  5  0  0  A¥o  =  t%%Vo   or   .32500. 

6.  Reduce  .45  to  ten-thousandths. 

7.  Reduce  6.5  to  thousandths. 

8.  Reduce  23  to  hundredths. 

9.  Reduce  62.5  to  ten-thousandths. 

10.  Reduce  .048  to  hundred-thousandths. 

11.  Reduce  406.062  to  millionths. 

12.  Reduce  .4500  to  hundredths. 

PROCESS :  .4  5  0  0  =  .4  5,  Ans. 

13.  Reduce  .5000  to  tenths. 

14.  Reduce  2.4000  to  hundredths. 

Art.  92.  Principles. — 1.  Annexing  ciphers  to  a  deci- 
mal^ or  decimal  ciphers  to  an  ititeger,  does  not  change 
its  value.     (Art.  GS.) 

2.  Removing  ciphers  from  the  right  of  a  decimal^  or 
decimal  ciphers  from  the  right  of  an  integer^  does  not 
change  its  value.     (Art.  63.)  * 

Case  II.—  Decimals  Reduced  to  Common  Fractions. 

1.  How  many  fiftlis  in  y\?     In  ^?     In  .6? 

2.  How  many  fourths  in  ^25^?     In  y%\?     In  .75? 

WRITTEN  EXERCISES. 

3.  Reduce  .225  to  a  common  fraction  in  its  lowest 
terms. 

PROCESS  :  .225  =  ^^  --=  4% J  -4n6'.     (Art.  64.) 


DECIMAL  FRACTIONS.  121 

4.  Eediice  .75  to  a  common    fraction   in   its   lowest 


terms. 

5.  .625. 

8. 

.075. 

11. 

.0096. 

14. 

.0032. 

G.  .0625. 

9. 

.024. 

12. 

3.25. 

15. 

12.375. 

7.  .125. 

10. 

.512. 

13. 

21.075. 

16. 

25.032. 

Art.  93.  EuLE. — To  reduce  a  decimal  to  a  common 
fraction  in  its  lowest  terms,  Omit  the  decwial  point  and 
supply  the  denominator^  and  then  reduce  the  fraction  to 
its  lowest  terms. 

Case  III. — Common  Fractions  Reduced  to  Decimals. 

1.  How  many  tenths  in   i?     In  ^?     |?     |? 

2.  How  many  hundredths  in  2^^     ^^^  ^\  ^     A^ 

WKITTEN   EXERCISES. 

3.  Reduce  ^^  to  a  decimal. 

25)3.00(.12,  Arts. 

2  5  Since  ^\  =  ^^  ^^  ^>  ^"d 

process:          -— t  3  =  3.00    (Art.   92),  ^\  = 

^^  ^V  of  3.00  =.12. 

4.  Reduce  |  to  a  decimal. 

6.^.  9.^.  12.  3,-V-  15-  12A. 

7.1.  10.3%.  13.  21|.  16.25^-. 

17.  Reduce  .621  to  thousandths. 

18.  Reduce  .012|  to  ten-thousandths. 

19.  Reduce  12.06L  to  millionths. 

Art.  94.  Rule. — To  reduce  a  common  fraction  to  a 
decimal,  Annex  decimal  ciphers  to  the  numerator  and 
divide  by  the  denominator,  and  point  off  as  many  deci- 
mal places  in  the  quotient  as  there  are  annexed  ciphers. 


122  INTERMEDIATE   ARITHMETIC. 

LESSON    III. 

^dddilion  of  decimals, 

1.  What  is  the  sum  of  5  tenths  and  4  tenths?  6 
tenths  and  9  tenths? 

Solution.  —  6  tenths  and  9  tenths  are  15  tenths,  which 
is  equal  to  1  unit  and  5  tenths. 

2.  What  is  the  sum  of  8  hundredths  and  7  hun- 
dredths?    18  hundredths  and  7  hundredths? 

3.  Wliat  is  the  sum  of  28  thousandtlis  and  9  thou- 
sandths?    56  tliousandths  and  22  thousandths? 

WRITTEN  EXERCISES. 

4.  What  is  the  sum  of  24.6,  307.08,  93.609,  .456, 
400.06,    37.027. 

PROCESS.  Since  only  like  orders  can  be  added, 

2  4.6  write  the  figures  of  the  same  order  in 
307.08  the  same  column.     Since  ten  units  of 

9  3.6  09  any   order   make   1    unit   of    the    next 

.4  56  left-hand  order,  begin  at  the  right,  and 

400.06  add  as  in   simple  numbers.     Place  the 

3  7.0  2  7  decimal  point  between  units  and  tenths 
86  2.8  3  2,  Alls,  in  the  amount. 

5.  What  is  the  sum  of  .4506,  .709,  and  27.0508? 

6.  Add  15.34,  6.078,  60.804,  and  99.875. 

7.  Add  $21.94,  $87,075,  $9,858,  and  $807,621. 

8.  Add  thirty-nine  hundredths,  six  hundred  and 
eight  ten-thousandths,  and  eighty-seven  thousandths. 

9.  What  is  the  sum  of  47.6  miles,  19.48  miles, 
34.75   miles,   and   76.625   miles? 

Art.  95.  Rule. — To  add  decimals,  1.  Write  the  num- 
bers so  that  figures  of  the  same  order  shall  be  in  the  same 
column. 


DECIMAL  FRACTIONS.  123 

2.  Beginning  at  the  rights  add  as  in  the  addition  of 
integers^  and  place  the  decimal  point  at  the  left  of  the 
tenths'  order  in  the  amount. 


LESSON    IV. 
Suhtracti07i  q/'  Decimals. 

1.  From  8  tenths  take  5  tenths. 

2.  From  5  tenths  take  5  hundredths. 

Solution. — 5  tenths  are  equal  to  50  hundredths,  and  50 
hundredtlis  less  5  hundredths  are  45  hundredths. 

3.  From  7  hundredths  take  4  thousandths. 

WRITTEN    EXERCISES. 

4.  From  56.403  take  18.6. 

Since  only  like  units  can 

5  6^4  0  3  be    subtracted,     write     the 

process:    1  8.6  numbers  so  that  figures  of 

3  7.803,  Ans.  tlie  same  order  shall  be  in 

the  same  column.    Since  ten 

units  of  any  decimal  order 

5.  From  56.6  take  18.403.    make  one  unit  of  the  next 

left-hand  order,  subtract  as 
"^"•"^^  in   integers,   and    place    the 

process:.  18.403  decimal  point  at  the  left  of 

3  8.197,  Ans.  the  tenths'  order  in  the  re- 

mainder. 

6.  From  45.3  take  28.756. 

7.  From  .0407  take  .008075. 

8.  From  twelve  thousandths  take  eight  millionths. 

9.  From  eight  tenths  take  eight  ten-thousandths. 

10.  From  47.065  +  36.87  take  9.08  +  43.375. 

11.  From   the   sum   of  twenty-five   thousandths   and 
forty-six  ten-thousandths  take  their  difference. 


124  INTERMEDIATE  ARITHMETIC. 

Art.  96.  EuLE.  — To  subtract  decimals,  1.  Write  the 
numbers  so  that  figures  of  the  same  order  shall  he  in  the  same 
column. 

2.  Subtract  as  in  the  subtraction  of  integers,  and  place 
the  decimal  point  at  the  left  of  the  tentlis'  order  in  the  re- 
mainder, 

LESSON   V. 
Mnltiplicatio7i  of  Decimals. 

1.  How  much  is  3  times  2  tenths?     3  times  ^? 

2.  How  much  is  4  times  3  tenths?    4  times  .4? 

3.  How  much  is  5  times  jf^?     5  times  .05? 

4.  How  much  is  ^2^  x  i^?     .2  X  -3?     .4  X  -7? 

5.  How  much  is  yV  X  yw?     .1  X  .05?     .4  X  -06? 

6.  How  much  is  ^^  x  yf^?     .02  X  .05. 

7.  How  much  is  y|^  X  y^W?     -06  X  -008? 

8.  What  is  the  denominator  of  the  product  when 
tenths  are  multij^lied  by  units?     Tenths  by  tenths? 

9.  What  decimal  order  is  produced  when  tenths 
are  multiplied  by  hundredths?  Hundredths  by  hun- 
dredths?    Hundredths  by  thousandths? 

10.  What  is  the  number  of  decimal  orders  in  the 
jDroduct  of  any  two  decimals? 

W^RITTEN   EXERCISES. 

11.  Multiply  .435  by  .65. 

PROCESS. 

435  Since  thousandths  multiplied  by 

55  hundredths    produce    hundred- thou' 

^TWT~  sandthsy   the   product   of   .435    and 

i)(*^f)  -65  contsLUiQ  five  decimal  orders,  or 

as  many  as  both  of  the  factors. 


.2  82  7  5,  Ans, 


DECIMAL  FRACTIONS.  125 

12.  Multiply  .347  by  .73.  19.  30.3  by  .044. 

13.  Multiply  .48  by  .36.  20.  .008  by  .007. 

14.  Multiply  .067  by  6.5.  21.  .075  by  .48f 

15.  3.42  by  .054.  22.  2.42  by  50. 

16.  47.5  by  3.4."  23.  .0075  by  2.8. 

17.  492  by  3.06.  24.  43.6  by  .073. 

18.  650  by  .24.  25.  .024  by  .06i 

26.  Multiply  4.35  by  10.     By  100. 

PROCESS.  Since   the  value  of  decimal  or- 

ders increases  from  right  to  left  ten- 
4.35  X  10  ==43.5,    Ans.   ^^^^  ^j^^^    ^^^  p^.    2)^   the   removal 

4.35  X  100  —  435,^^5.  of  the  decimal  point  one  place  to 
the  right  removes  each  figure  one 
order  to  the  left,  and  hence  multiplies  4.35  by  10.  The  re- 
moval of  the  decimal  point  two  places  to  the  right  multiplies 
4.35  by  100. 

27.  Multiply  4.085  by  100.     By  1000. 

28.  Multiply  3.0048  by  1000.     By  100000. 

PRINCIPLES  AND  EULES. 

Art.  97.  Principles.  —  1.  The  number  of  decimd  pJmes 
in  the  product  equals  the  number  of  decimal  places  in  both 
factors, 

2.  Each  removal  of  the  decimal  point  one  place  to  the  right 
multiplies  a  decimal  by  10. 

Art.  98.  Rules. — 1.  To  multiply  one  decimal  by 
another,  Multiply  as  in  the  multiplication  of  integers^  and 
point  off  as  many  decimal  places  in  the  product  as  there  are 
decimal  places  in  both  multij)licand  and  midtiplier. 

Note. — If  there  be  not  enough  decimal  figures  in  the  product, 
supply  the  deficiency  by  prefixing  decimal  ciphers. 

2.   To    multiply   a    decimal    by   10,   100,   1000,   etc., 


126  INTERMEDIATE  ARITHMETIC. 

Remove  the  deciinal  point  as  many  places  to  tlie  rigid  as  tJwre 
are  dplwrs  in  the  multiplier. 

Note. — If  there  be  not  enough  decimal  places  in  the   product, 
supply  the  deficiency  by  annexing  ciphers. 


LESSON    VI. 
Dlvisio7i  of  decimals, 

1.  How  many   times   are   2   tenths   contained  in    8 
tenths?     3  tenths  in  9  tenths? 

2.  How    many    times    are    4    hundredths    contained 
in  12  hundredths?     6  hundredths  in  42  hundredths? 

3.  How  much  is  ^^  -^  t%-     ^  tenths  ~  3  tenths? 

4.  How  much  is  .06  -^  .02?     .56  ^  .07? 

5.  Of  what  order  is  the   quotient  when  tenths  are 
divided  by  tenths?     Hundredths  by  hundredths? 

6.  Of  what  order  is  the  quotient  when  any  number 
is  divided  by  a  lii<:e  number? 

WRITTEN  EXERCISES. 

7.  Divide  6.25  by  .25. 

PROCESS. 

.2  5)62  5(25    Ans.  Since  25  hundredths  are   con- 

f^Q  tained  in  625  hundredths  (a  like 

number)   25   times,  the  quotient 
is  an  integer. 


125 
125 


8.  Divide  .625  by  .25. 

PROCESS.  Since  the  divisor  (.25)  and  the 

.2 5 ).6 2 5(2.5,^1/15.  first    partial    dividend    (.62)   are 

50  both  hundredths  (like  numbers), 

125  the  first  quotient  figure  is  units^ 

12  5  and  hence  the  second  is  tenths. 


DECIMAL   FRACTIONS.  127 

9.  Divide  17.28  by  .48.     By  4.8. 

10.  Divide  3.528  by  .042.     By  12.6. 

11.  Divide  .9408  by  8.4.     By  .084. 

12.  Divide  .06241  by  79.     By  .079. 

13.  Divide  18.816  by  1.68.     By  168. 

14.  Divide  $17,595  by  $.85.     By  $2.07. 

15.  Divide  .0768  by  9.6.     By  .096. 

16.  Divide  62.5  by  .025. 

PROCESS. 

.025)62.500(2500,^/15.  %   annexing    two    decimal 

50  ciphers  to  62.5,  the  dividend 

T^T  and    divisor    are    made    like 

^  rt  c  numbers,  and  hence  their  quo- 

tient  is  an  integer. 

00 

17.  Divide  25.6  by  .032.     By  .16. 

18.  Divide  2.5  by  1.25.     By  .0125. 

19.  Divide  45.3  by  3.02.     By  .0302. 

20.  Divide  80.5  by  .35.     By  .00035. 

21.  Divide  402.5  by  1.75.     By  .0175. 

22.  Divide  34.5  by  10.     By  100. 

PROCESS.  The    removal    of   the    decimal 

8  4.5^-10=3.4  5.  point  one  order   to   the   left,   re- 

345-^100=  345        moves    each   figure    in   34.5   one 

order    to    the    right,    and    hence 

divides  its  value  by  10;   and  the  removal  of  the  decimal   point 

two  places  to  the  left  divides  it  by  100. 

23.  Divide  436.7  by  100.     By  1000. 

24.  Divide  234.6  by  1000.     By  100000. 

PRINCIPLES  AND  RULES. 

Art.  99.  Principles. — 1.  The  dividend  contaim  as  many 
decimal  places  as  both  divisor  and  qnoti£nt. 


128  INTERMEDIATE  ARITHMETIC 

2.  Tlie  quotient  contains  as  many  decimal  places  as  the 
number  of  decimal  places  in  the  divideml  exceeds  tlie  number 
in  Hie  divisor. 

3.  Each  removal  of  tJw  decimal  point  one  place  to  ilie  left 
divides  a  decimal  by  10. 

Art.  100.  EuLES. — 1.  To  divide  one  decimal  by  an- 
other, Divide  as  in  the  division  of  integers j  and  point  off  as 
many  de<iimal  places  in  the  quotient  as  the  nuynber  of  deci- 
mal places  in  tJie  dividend  exceeds  the  number  in  the  divisor. 

Notes. — 1.  When  tlie  divisor  contains  more  decimal  places  than 
the  dividend,  supply  the  deficiency  in  the  dividend  by  annexing 
decimal  ciphers. 

2.  When  the  quotient  has  not  enough  decimal  figures,  supply 
the  deficiency  by  prejixlng  decimal  ciphers. 

3.  When  there  is  a  remainder,  the  division  may  be  continued 
by  annexing  ciphers,  each  cipher  thus  annexed  adding  one  deci- 
mal place  to  the  dividend.  Sufficient  accuracy  is  usually  secured 
by  carrying  the  division  to  three  decimal  places. 

2.  To  divide  a  decimal  by  10,  100,  1000,  etc., 
Remove  the  decimal  point  as  many  places  to  the  left  as  there 
are  ciphers  fa  the  divisor. 


REVIEW   PROBLEMS. 

1.  Express  decimally  i^  of  one  hundredth. 

2.  Eeduce  -^^^-^  to  a  decimal. 

3.  Change  .0325  to  a  common  fraction. 

4.  Divide  84.564  by  9.72. 

5.  Divide  36.72  by  3.6.     By  .036. 

6.  Divide  25.6  X   56  by  .0128. 

7.  Divide  348.6  by  100.     By  2000. 

8.  Divide  17.28  by  24.     By  2400. 

9.  Multiply  27.5  -r-  .025  by  76.8  --  .48. 

10.  What   will    be    the    cost    of   excavating    437.24 
cubic  yards  of  earth  at  $1.65  a  cubic  yard? 


SF.CTION  IX. 

UJVITi;'D   STATICS  MOJV^T. 


LESSON    I. 

Art.  101.  United  States  Money  is  the  legal  cur- 
rency of  the  United  States.  It  is  also  called  Federal 
Money. 

The  denominations  are  mills,  cents ^  dimes,  and 
dollars. 

Table. 

10    mills  (;??.)  .     .     are  1  cent       .     .     <?.  or  cL 
10    cents      .     .     .     are  1  dime      .     .     d. 
10    dimes     .     .     .     are  1  dollar     .     .     $. 

$1-  lOf?.  =-  100  c.  -  1000  ^«. 

I.  A.— 9.  (129) 


130  INTERMEDIATE   ARITHMETIC. 

Notes.  —  1.  United  States  money  consists  of  Coin  and  Paper 
Money.  Coin  is  called  Specie  Currency,  or  Specie,  and  paper  money 
is  cailed  Paper  Currency. 

2.  The  principal  gold  coins  are  the  double  eagle  ($20),  eagle 
($10),  half-eagle  ($5),  quarter-eagle  ($2^),  three-dollar  piece,  and 
dollar. 

The  silver  coins  are  the  dollar,  half-dollar,  quarter-dollar,  and 

dime.  The  smaller  coins  are  the  five-cent  piece,  three-cent  piece, 
two-cent  piece,  and  cent.  The  five-cent  piece  and  the  three-cent 
piece  are  made  of  copper  and  nickel,  and  the  two-cent  piece  and 
cent  are  made  of  bronze,  an  alloy  of  copper,  tin,  and  zinc. 

3.  Paper  money  consists  of  notes  issued  by  the  United  States, 
called  Treasury  Notes,  and  bank  notes  issued  by  banks. 

1.  How    many    mills    in    1    cent?      In    5    cents? 

4  cents?     7   cents?     9    cents? 

2.  How    many    cents    in    1    dime?      In    4    dimes? 

5  dimes?      8   dimes?     10   dimes? 

3.  How  many  dimes   in    1    dollar?     In  3  dollars? 
G  dollars?     4  dollars?     8  dollars? 

4.  How   many   cents   in    10    mills?     In   50   mills? 
40  mills?     60  mills?     80  mills? 

5.  How  many  dimes   in   40   cents?     In    90   cents? 
GO  cents?     70  cents?     100  cents? 

G.  How  many  dollars  in  50  dimes?     In  70  dimes? 
60  dimes?     80  dimes?     100  dimes? 

7.  How  many  dimes  in   25   cents?     In    75   cents? 
15  cents?     35  cents?     95  cents? 

8.  How   many   cents    in    35    mills?     lu    65   mills? 
25  mills?     75  mills?     95  mills? 

9.  How   many  cents    in   5   dimes?     In    15    dimes? 
45  dimes?     30  dimes? 

10.  How  many  dimes  in  15  cents?  In  95  cents? 
85  cents?     65  cents? 

11.  How  many  cents  in  1  dollar?  In  5  dollars? 
7  dollars?     9  dollars?     8  dollars? 

12.  How  many  dollars  in  200  cents?  In  500  cents? 
400  cents?     900  cents? 


itnitp:d  states  :money.  131 


WRITTEN  EXERCISES. 

Art.  102.  Accounts  are  kept  in  dollars  and  cents: 
mills  are  used  only  in  making*  calculations.  The 
figures  denoting  dollars  are  preceded  by  the  sign,  $. 
The  first  two  figures  at  the  right  of  dollars  denote 
cents,  and  the  third  figure  denotes  mills.  The  figures 
denoting  dollars  are  separated  from  those  denoting 
cents  by  a  period  (.),  called  a  Separatrix.  Thus, 
$45,307   is   read   45   dollars,   30   cents,   7   mills. 

Note.— Tlie  pupil  should  here  be  tanglit  that  the  first  figure  at 
tlie  right  of  the  separatrix  denotes  tenths  oi  a  dollar;  the  second, 
hundredtlis ;  the  third,  thousandths.  He  should  also  be  taught  to 
read  the  following  numbers  decimally,  and  to  write  similar  num- 
bers, when  dictated,  decimally. 

Copy  and  read  the  following: 

(13)  (14)  (15)  (16) 

$3.4  5  $0,0  7  5  $40,04  5  $10  0. 

$3,506  $0,0  0  5  $15.15  $405. 

$1,05  5  $3.0  8  $10,015  $704.50 

$0.7  5  $9,0  0  9  $6  0.6  0  $8  00.08 

17.  Write  in  figures  4  dollars  40  cents. 

18.  Write  12  dollars  33  cents  5  mills. 

19.  Write  60  dollars  6  cents  4  mills. 

20.  Write  75  cents  5  mills. 

21.  Write  30  cents;   40  cents  7  mills. 

22.  Write  300  dollars  3  cents  7  mills. 

23.  Write  500  dollars  5  mills. 

24.  Write  25  cents  7  mills;    6  cents  1  mill. 

25.  Write  10  dollars  3  cents  8  mills. 

26.  Write  1000  dollars;    50  dollars  5  cents. 

27.  Write  25  dollars  1  cent  5  mills. 

28.  Write  500  dollars  3  mills. 

29.  Write  5  mills;    5  cents  5  mills. 

30.  Write  60  dollars  60  cents  6  mills. 


1B2 


INTERMEDIATE   ARITHMETIC. 


1. 
2. 

8. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
1(3. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 


LESSON   II. 

^ediiciion  of  United  States  Money. 

MENTAL  AND  WRITTEN    EXERCISES. 

How  many  cents  in  $15? 

How  many  mills  in  $15? 

How  many  cents  in  $5.25? 

How  many  mills  in  $1,375? 

How  many  mills  in  $.62i? 

Keduce  $75  to  cents;    $108  to  cents. 

Eeduce  $125  to  cents;    $230  to  cents. 

Eeduce  $12.65  to  cents;    $5.60  to  cents. 

Eeduce  $1.08  to  cents;    $8.01  to  cents. 

Eeduce  $25  to  mills;    $40  to  mills. 


Ans. 

15000m. 

Ans, 

525c. 

Ans. 

1375/n. 

Am. 

625m. 

Eeduce  $.12|-  to  mills; 


$3,121  to  mills. 


Eeduce  $.375  to  mills;    $.105  to  mills. 
Eeduce  $4.50  to  mills;    $3.03  to  mills. 
Eeduce  $.45  to  mills;    $.05  to  mills. 
Eeduce  $102  'o  cents;    $10  to  cents. 
Eeduce  $120  to  mills;    $45  to  mills. 
Eeduce  $.25  to  cents;    $.01  to  cents. 
How  many  dollars  in  7500  cents?     Ans. 
How  many  dollars  in  7550  cents?    Ans. 
How  many  dollars  in  3125  mills?    Ans. 
How  many  dollars  in  4000  mills? 
Eeduce  1507  cents  to  dollars. 
Eeduce  1001  cents  to  dollars. 
Eeduce  1500  mills  to  dollars. 
Eeduee  10250  mills  to  dollars. 
Eeduce  5000  cents  to  dollars. 
Eeduce  5000  mills  to  dollars. 
Eeduce  375  cents  to  dollars. 
Eeduce  375  mills  to  dollars. 


$75. 

$75.50. 

$3,125. 


UNITED  STATES  MONEY.  133 

30.  Ecduce  $4.50  to  mills. 

31.  Eeduce  4500  mills  to  dollars. 

32.  Ecduce  $10.10  to  cents. 

33.  Eeduce  1010  mills  to  dollars. 

Art.  103.  EuLES.  —  1.  To  reduce  dollars  to  cents, 
Annex  two  ciphers, 

2.  To  reduce  dollars  to  mills,  Annex  three  ciphers. 

3.  To  reduce  cents  to  mills,  Annex  one  cipher. 

4.  To  reduce  dollars  and  cents  to  cents,  or  dollars, 
cents,  and  mills  to  mills.  Remove  the  separatrix  and  the 
dollar  sign. 

5.  To  reduce  cents  to  dollars,  Place  the  separatrix  be- 
fore the  second  right-hand  figure^  and  prefix  the  dollar 
sign. 

6.  To  reduce  mills  to  dollars,  Place  the  separatrix  be- 
fore the  third  right-hand  figure^  and  prefix  the  dollar  sign. 

Note.— Annexing  two  ciphers  is  multiplying  by  100,  aiTd  point- 
ing otf  two  figures  from  the  right  is  dividing  by  100.  (See  Arts. 
37,  45.) 

LESSON    III. 
^ddi/ion  and  Subtraction, 

AVRITTEN   EXERCISES. 

1.  What    is    the    sum    of   $50,    $16.50,    $3,333,    and 

$.87i? 


PROCESS. 

$5  0. 
16.50 


Write  the  several  numbers  to  be  added 
so  that  units  of  the  same  denomination 
may  stand  in  the  same  column,  and  then 
add,    as    in    simple    numbers.     The    dollar 

-  '  sign  need    be  written   but   once. 

$7  0,7  0  8,  Ars. 

2.  What  is  the  sum  of  $1.20,  $5,  $10.15,  $.85,  and 
$.621? 


134  INTERMEDIATE   ARITHMETIC. 

3.  What  is  the  buiu  of  ^1),  $12.50,  $4,371-,  $40.08, 
$6.33,  and  $25.? 

4.  Add  $45,371  $100.50,  $16,121,  $37,  $9.05,  $.87^, 
$4.44,  and  $95. 

5.  From  $37.50  take  $5.(;2i 

niocEss.  6.  From  $6.37^  take  $5.87|. 

$87,500  7.  From  $100  take  $1,256. 

5.6  2  5  8.  From  $10  take  $.10. 

$31,8  75,  Ans.  9.  A    man    sold    a    carriage    for 

$160.75,  a  horse  for  $125,  a  set 
of  harness  for  $26,371,  and  a  saddle  lor  $15,621; 
what  was   the    amount    received? 

10.  A  grocer  buys  flour  at  $8,621  a  barrel,  and 
sells  it  at  $10  a  barrel:    wdiat  is  his  gain? 

11.  A  merchant  paid  $32.50  for  a  barrel  of  sugar, 
and  sold  it  for  $35:    how  much  did  he  gain? 

12.  A  laborer  earns  $17.50  a  week,  and  his  expenses 
are  $12,621  a  week :  how  much  can  he  save  each 
week  ? 

13.  A  man  bought  a  house  and  lot  for  $3506.75, 
and  sold  it  for  $4000:    what  was  his  gain? 

14.  A  man  bought  a  carriage  for  $160,  2)aid  $22.75 
for  repairing  it,  and  then  sold  it  for  $180:  how  much 
did  he  lose? 

15.  Mr.  Smith  bought  a  house  and  lot  for  $4500, 
paid  $40.50  for  a  fence,  $105.65  for  painting,  $47.12 
for  papering,  and  $25  for  other  improvements:  what 
will  he  make  if  he  sell  the  property  for  $5000? 

Art.  104.  EuLE. — To  add  or  subtract  sums  of  money. 
Write  units  of  the  same  denomination  in  the  same  column, 
add  or  subtract  as  in  simple  numbers,  and  separate  dollars 
and  cents  by  a  period,  and  prefix  the  dollar  sign. 


UNITED  STATES  MONEY.  135 

LESSON    IV. 

Multq^licailon  and  Division, 

WRITTEN  EXERCISES. 

1.  What  will  9  cords  of  wood  cost  at  $3,621  a  cord? 

PROCESS. 

$3,6  2  5  If  1    cord    of  wood    cost    $3,625,  9   cords 

9  will  cost  9  times   $3,625,  which  is  $32,625. 

$32,6  2  5,    A)is. 

2.  What   will    16    barrels    of   flour    cost   at   $7.50    a 
barrel  ? 

3.  What   will    40    yards    of  cloth    cost   at   $1.12^  a 
yard? 

4.  What  will  12|  tons  of  hay  cost  at  $11.50  a  ton? 

5.  At    18|-   cents    a   dozen,   what    will    12    dozen   of 
eggs  cost? 

6.  If  a  boy  earn  $4.37-|-  a  week,  how  much  will  he 
earn  in  20  weeks? 

7.  A  drover  sold   36   cows  at  $33.33^  a  head :    how 
much  did  he  receive  for  them? 

8.  What  will  90  bushels  of  wheat  cost  at  $1.62|^  a 
bushel? 

9.  If  9  cords  of  wood  cost  $32.62^,  what  will  1  cord 
cost? 

^^^>CESS.  If  9  (,Qrds  cost  $32,625,  1  cord  will 

9  )  $3  2.6  2  5  cost  \  of  $32,625,  which  is  $3,625,  or 

$3,6  2  5,  ^n^.  $3.62^. 

10.  If  12   pounds   of  sugar  cost  $2.16,  what  will  1 
pound  cost? 

11.  A    man    paid   $1687.50    for    45    acres    of   land: 
what  was  the  price  an  acre? 


136  INTERMEDIATE    ARITHMETIC. 

12.  A  grocer  paid  $135  for  18  barrels  of  flour: 
what  was  the  cost  a  barrel? 

13.  A  man  earned  $91  in  8  weeks:  how  much  did 
he  earn  a  week  ? 

14.  At  $.12^  a  dozen,  how  many  dozen  of  eggs  can 
be  bought  for  $5? 

PROCESS.  $5  ==5000  mills,   and   $.12} 

125w.)5000m.(40,^ln5.  ==125    mills.       Hence,     $o  ~- 

500  $.12}  =  5000  mills --  125  mills, 

0  which  is  40. 

Note. — When  both  divisor  and  dividend  are  denominate  num- 
bers, they  must  be  reduced  to  the  same  denomination  before  di- 
viding. 

15.  At  $1.25  a  bushel,  how  many  bushels  of  corn 
can  be  bought  for  $75? 

16.  At  31  cents  apiece,  how  many  lemons  can  be 
bought  for  $7? 

17.  If  a  boy  earn  75  cents  a  day,  in  how  many 
days  will  he  earn  $24? 

18.  At  37^  cents  a  bushel,  how  many  bushels  of 
oats  can  be  bought  for  $57.75? 

19.  A  farmer  sold  35  pounds  of  butter  at  20  cents 
a  pound,  and  received  in  payment  muslin  at  12^ 
cents  a  yard:  how  many  yards  of  muslin  did  ho 
receive? 

20.  A  farmer  exchanged  16  cows,  at  $27.50  a  head, 
for  sheep,  at  $5.50  a  head:  how  many  sheep  did  he 
receive  ? 

21.  How  many  lemons,  at  2^  cents  each,  can  be 
bought  for  20  oranges,  at  5  cents  each? 

22.  Multiply  $12.62^  by  15,  and  divide  the  product 
by  $2,525. 

23.  Multiply  $1.25  by  18,  and  divide  the  product 
by  $.62f 


TNITED   STATES   MONEY.  137 

24.   Multiply  $5.75  by  25,   and    divide   the    product 

by  $.571 

Art.  105.  EuLES. — 1.  To  multiply  or  divide  sums  of 
money  by  an  abstriict  immber,  Multiply  or  divide  as  in 
diiiple  numbers,  separate  dollars  ami  cents  in  the  result  by  a 
j)eriody  and  prefix  the  dollar  sign. 

2.  To  divide  one  sum  of  money  by  another,  Reduce 
both  numbers  to  the  same  denomination,  and  divide  as  in 
simple  numbers, 

LESSON    V. 
M  iscellaneoiis    Written   ^JProblenis, 

1.  What  is  the  sum  of  $13.45,  $9.87,  $100,  $.87, 
$1.40,  and  $14? 

2.  From  $10  take  5  mills. 

3.  From  $500  take  500  cents. 

4.  Multiply  $15,331-  by  33. 

5.  Divide  $50  by  50  cents. 

6.  A  man's  income  tax  in  1868  was  $55.75,  his 
State  and  city  tax  $68.35,  and  his  other  taxes  $7.50 : 
what  was  the  amount  of  his  taxes? 

7.  A  man  bought  a  house  and  lot  for  $5400,  and, 
after  expending  $1500  for  improvements,  sold  the 
property  for  $7500 :    how  much  did  he  gain  ? 

8.  What  will  60  ];)Ounds  of  butter  cost  at  331^  cents 
a  pound? 

9.  What  is  the  cost  of  35  reams  of  paper,  weighing 
44  pounds  each,  at  18  cents  a  pound? 

10.  How  many  yards  of  carpeting,  at  $1.75  a  yard, 
can  be  bought  for  $350? 

11.  A  fruit  dealer  makes  a  net  profit  of  20  cents 
on  each  bushel  of  apples  he  sells :  how  many  bushels 
must  he  sell  to  make  $80? 


138  INTERMEDIATE   ARITHMETIC. 

12.  A  widow  is  to  receive  one  third  of  an  estate 
of  $12000,  and  the  remainder  is  to  be  divided  equally 
between  5  children  :  what  is  tlie  share  of  each  child  ? 

±3.  A  fruit  dealer  sold  144  baskets  of  peaches  for 
S252:    what  was  the  price  per  basket? 

14.  If  40  acres  of  land  cost  $1400,  how  many  acres 
can  be  bought  for  $1750? 

15.  A  man  sold  15  cords  of  wood  at  $4.50  a  cord, 
and  received  in  payment  10  barrels  of  flour:  what 
did  the  flour  cost  him  a  barrel? 

16.  If  8  barrels  of  salt  cost  $36,  what  will  13 
barrels    cost? 

17.  A  grain  dealer  bought  15000  bushels  of  wheat 
at  $1.35  a  bushel,  and  sold  it  the  next  week  for 
$1.48  a  bushel:    what  was  his  gain? 

18.  A  workman  receives  $1.50  a  day,  and  his  living 
costs  him  $.75  a  day:  how  much  can  he  lay  up  in  a 
year,  if  he  work  310  days? 

19.  A  drover  bought  240  sheep  at  $4.50  a  head, 
drove  them  to  market  at  an  expense  of  $75,  and 
then  sold  them  at  $6.50  a  head:  how  much  did  he 
make  ? 

20.  A  farmer  exchanged  40  pounds  of  butter  at 
22  cents  a  pound,  and  8  dozen  of  eggs  at  12^  cents 
a  dozen,  for  cotton  cloth  at  10  cents  a  yard :  how 
many  yards  of  cloth  did  he  receive? 

21.  A  farmer  exchanged  8  cows  valued  at  $37.50  a 
head,  for  sheep  valued  at  $7.50  a  head:  how  many 
sheep  did  he  receive? 

22.  If  a  boy  pays  $2.50  a  hundred  for  papers,  and 
sells  them  at  5  cents  apiece,  how  much  does  he  make 
on  100  papers? 

23.  A  farmer  sold,  one  year,  200  bushels  of  wheat, 
at   $1.80   a  bushel;    500   bushels   of  corn,   at  $1.15  a 


UNITED  STATES  MONEY.  139 

bushel ;  65  bushels  of  potutoep.,  at  80  cents  ii  bushel ; 
12  tons  of  hay,  at  $16.50  a  ton;  and  225  pounds  of 
butter,  at  20  cents  a  pound:  what  was  the  amount 
of  his  annual  product? 

24.  A  man  bought  250  bushels  of  coal,  at  15  cents 
a  buslicl;  7  cords  of  wood,  at  $5.50  a  cord;  18 
bushels  of  potatoes,  at  $.90  a  bushel ;  and  9  barrels 
of  apples,  at  $2.75  a  barrel :  how  much  did  he  pay 
for  all? 

25.  A  bookseller  sold  12  geographies,  at  $1.75;  20 
readers,  at  $.85;  30  arithmetics,  at  $.65;  and  45 
sj^ellers,  at  $.30 :    what  was  the  amount  of  the  bill  ? 

26.  The  annual  expenses  of  a  man's  family  are 
as  follows:  provisions,  $350;  clothing,  $400;  fuel, 
$95;  books  and  periodicals,  $50;  house-rent,  $240; 
and  all  other  expenses,  $150:  if  he  receive  an  annual 
salary  of  $1500,  how  much  can  ho  lay  up  each  year? 

To  Teachers.  —  See  Manual  of  Arithmetic  for  addi- 
tional review  problems  for  dictation. 

LESSON    VI. 

:bizzs. 

L  Columbus,  0.,  June,  10,  1869. 

Mr.  Charles  Wilson 

Bought  of  James  Cooper  &  Co. : 

Vi  lbs.  Coffee,  @  30c $3.90 

4  lbs.  Butter,  @  35c 1.40 

10  lbs.  B'k't  Flour,  @     6c 60 

12  lbs.  Dried  Beef,   @  24c 2.88 

25  lbs.  Sugar,  @  18c. 4.50 

3  lbs.  Starch,  @  20c ^ ^ 

$13.88 
Received  payment^ 

James  Cooper  &  Co. 


140  INTERMEDIATE  ARITHMETIC. 

2.  Chicago,  Jan.  3,  1869. 

Joseph  Masox 

Bought  of  Peter  &  Brothers: 

27  yds.  Brussels  carpeting,  @  $2.60 

23  yds.  Ingrain         ^*             @  1.75 

8|  yds.  Oil  Cloth,                 @  1.20 

32  yds.  Curtains,                    @  .60 


$ 
Received  payment^ 

Peter  &  Brothers, 

Per  Smith. 

What  is  the  amount  of  the  above  bill? 


3.  Nashville,  Tenn.,  Oct.  8,  1868. 

Samuel  Mills 

To  Jones,  Smith  &  Co.,  Dr. 

To    7  yds.  Broadcloth,  @  $6.50  ...... 

*'      3^  yds.  Doeskin,  @  2.75 

*'      7J  yds.  Linen,  @  .90 

*^      2-j  doz.  Handkerchiefs,  @  1.50 

*'  12^  yds.  Muslin,  @  .18  .     ....     . 

"      9  yds.       "     bleached,  @  .33 

"  12  yds.  Silk,  @  1.60 

"  19  yds.  Binding.  @  .08 


Becelved  imymenty 

Jones,  Smith  &  Co. 

"What  is  the  amount  of  the  above  bill? 


4. 


UNITED  STATES  MONEY.  HI 

St.  Louis,  May  23,  18(39. 


Henry  Williams 

IS(}9.  Bought  of  Isaac  Clarke: 


Mch.  10,    5  Pair  Calf  Boots, 


@  $5.75 


Ladies'  Gaiters,    @  3.10  . 

Children's  Shoes,  @  1.75  . 

Coarse  Boots,         @  2.75  . 

Calf  Shoes,  @  3.25  . 

Ladies'  Slippers,  @  1.20  . 

Calf  Boots,  @  5.75  .  

Received  payment ^ 

Isaac  Clarke. 

What  is  the  amount  of  the  above  bill  ? 


ii.          11 

8    " 

a          ic 

7     " 

Apr.     4, 

8     '^ 

a          a 

6     " 

<<          n 

7    *^ 

May  23, 

3     ^' 

Pittsburgh,  Pa.,  Dec.  15,  1868. 


Andrew  Wilson 


1868.  ^oi/^A^  o/ Smith  &  Waring  : 

July    5,  7  gross  Shirt  Buttons,  @  $4.50 

''        "  10  doz.    Linen  Napkins,  @     2.75 

Aug.  12,  8       "      Pair  Kid  Gloves,  @  12.50 

''  3^     '•       Linen  Handk'fs,  @     6.75 

''  4§     ''       Shirt  Bosoms,  @     6.00 

Dec.   15,  3|     ''      Silk  Gloves,  @     9.00 

''  8      ''      Pair  Socks,  @     5.50 


Received  payment. 

Smith  &  Waring. 

What  is  the  amount  of  the  above  bill? 


142  INTERMEDIATE   ARITHMETIC. 

6.  Indianapolis,  Ind.,  Aug.  18,  1876. 

Thos.  M.  Cochrane 

Bought  of  Jones,  Dunlap  &  Co. : 

12    doz.  Scythes,  @  $15  

12i    "      Scy.  Snaths,  @     16.50 

6      ''      Eakes,  @       2.25 

o     ''      Hoes,  @       5.75 

8|    ''      AVhetstones,  @       1.50 

$ 
Cr, 

June  20,  By  Cash $75.00 

Aug.     1,  By  2h  doz.  Scythes  returned     .     .     .     37.50 

$112.50 

Received  pmjmejit,  $ 

Jones,  Dunlap  &  Co. 


7.  Columbus,  O.,  July  1,  1876. 

Smith  &  Bell 

;i^gg^  In  account  with  George  Stationer. 


Feb. 

1, 

To     2J  M.  Envelopes, 

® 

$5.75 

li 

(( 

"      IJ  reams  Cap  Paper, 

@ 

8.00 

ti 

(( 

''      3     Blank  Books, 

@ 

1.25 

]\Ich. 

9, 

*'      5    doz.  Pencils, 

@ 

1.25 

(( 

*'    60     lbs.  Wrapping  Paper 

,  @ 

.10 

ti 

11 

"      6    vols.  Dickens, 
Cr. 

@ 

1.75 

$ 

Juno 

20, 

By  printing  1500  Circulars 

$5.50 

(i 

u 

By  printing  Letter  Heads 

. 

3.75 

11 

25, 

By  33  tokens  Press- work 

- 

16.50 

$ 

$ 

UNITED  STATES  MONEY.  143 


DEFINITIONS. 


Art.  106.  A  Bill  of  Goods  is  a  written  etatement 
of  goods  sold,  with  the  price  of  each  article  and  the 
entire  cost.  It  also  gives  the  date  and  place  of  the 
sale,  and  the  names  of  the  buyer  and  seller. 

A  bill  is  drawn  against  the  buyer,  or  Debtor,  and 
in  favor  of  the  seller,  or   Creditor. 

A  bill  is  receipted  by  writing  the  words  ^^  Received 
payment^''  at  the  bottom,  and  affixing  the  seller's 
name.  A  bill  may  be  receipted  by  a  clerk,  agent, 
or  an}^  other  authorized  person,  as  in  Bill  2. 

Art.  107.  When  sales  are  made  at  different  times, 
the  dates  may  bo  written  at  the  left,  as  in  Bills  4, 
5,  and  7. 

A  bill  presenting  a  debit  and  credit  account  between 
the  parties,  may  be  written  and  receipted  as  in  Bill  6. 


Questions  for  Eeview. 

What  is  United  States  money?  What  is  it  also  called? 
Of  what  does  United  States  money  consist?  AVhat  are  the 
principal  gold  coins?  Silver  coins?  What  are  the  lesser 
coins?  Of  what  metals  are  the  lesser  coins  made?  Name  the 
two  kinds  of  paper  money. 

What  are  the  principal  denominations  of  United  States 
money?  Kepeat  the  table.  In  what  denominations  are  ac- 
counts kept?  What  use  is  made  of  the  dollar  sign?  How 
are  dollars  and  cents  separated  ?  Is  the  separatrix  a  period 
or  a  comma?     Where  is  the  figure  denoting  mills  written? 

How  are  dollars  reduced  to  cents?  Dollars  to  mills? 
Cents  to  mills?  How  are  dollars  and  cents  reduced  to 
c^nts?  Dollars,  cents,  and  mills  to  mills?  How  are  cents 
reduced  to  dollars?    Mills  to  dollars? 


144 


INTERMEDIATE    ARITHMETIC. 


Give  the  rule  for  adding  or  subtracting  sums  of  money. 
Give  the  rule  for  multiplying  or  dividing  a  sum  of  money 
by  an  abstract  number.     By  another  sum  of  money. 

What  is  a  bill  of  goods?  What  does  it  contain?  Against 
whom  is  it  drawn  ?  How  is  a  bill  receipted  ?  By  whom  ? 
Where  are  the  dates  of  sales  written  ?    Items  of  credit  ? 


SECTIOlSr   X. 


LESSON    I. 

Art.  108.  Dry  Measure  is  used  in  measuring  grain, 
fruit,  most  vegetables,  coal,  and  many  other  dry 
articles. 

The  denominations  are  piiitSf  quarts,  pecks,  and 
'bushels. 


DENOMINATE  NUMBERS.  145 

Table. 

2  pints  (pt)      ,     .     are  1  quart     ,     .     .     .     qt. 

8  quarts   ....     are  1  peck      .     ,     ,     ,    pk. 

4  pecks     ....     are  1  bushel  ....     6m. 

1  bu.  -=  4  pk.  =  32  qt.  =  64  pt. 

Notes. — 1.  The  standard  bushel  is  ISJ  inches  in  diameter  and 
8  inches  deep.     It  contains  2150|  cubic  inches. 

2.  In  measuring  grain,  seeds,  and  small  fruits,  the  measure 
nuist  be  even  full ;  but  in  measuring  corn  in  the  ear,  potatoes, 
apples,  and  other  large  articles,  the  measure  must  be  heaping  full. 

1.  How  many  pints  in  3  quarts? 

Solution. — In  3  quarts  there  are  3  times  2  pints,  which 
is  6  pints. 

2.  How  many  pints    in    5    quarts?     In    8    quarts? 
In  10  quarts? 

3.  How  many  quarts  in  10  pints? 

Solution. — In  10  pints  there  are  as  many  quarts  as  2 
pints  are  contained  times  in  10  pints,  which  is  5  times. 

4.  How   many    quarts   in   8   pints?     In    14    pints? 
In  16  pints?     In  20  pints? 

5.  How  many  quarts   in    3   pecks?     In    5i   pecks? 
In  1\  pecks?     In  lOf  pecks? 

6.  How  many  pecks  in  16  quarts?     In  20  quarts? 
In  32  quarts?     In  56  quarts? 

7.  How  many  pecks  in  5  bushels?     In  7^  bushels? 
In  9f  bushels?     In  11  bushels? 

-  8.  How  many  bushels  in  12  peeks?  In  20  pecks? 
In  32  pecks?     In  40  pecks? 

9.  How  many  quarts  in  8  pecks?     In  12  pecks? 

10.  How  many  pints  in  8  quarts?     In  12  quarts? 

11.  What  part  of  a  quart  is  1  pint?     2  pints? 

12.  What  part  of  a  peck  is  1  quart?     3  quarts? 

13.  What  part  of  a  bushel  is  1  peck?  2  2:)ecks? 
3  pecks?     4  pecks?     5  pecks? 

I.  A.— 10. 


146  INTERMEDIATE  ARITHMETIC. 

14.  How  many  pecks  in  17  quarts?  In  27  quarts? 
In  33  quarts? 

15.  How  many  bushels  in  13  pecks?  In  23  pecks? 
In  33  pecks  ? 

16.  Wiiat  will  51^  quarts  of  plums  cost  at  4  cents 
a  pint? 

17.  A  man  carried  3|  pecks  of  cherries  to  market, 
and  sold  them  at  10  cents  a  quart:  how  much  did 
he  receive? 

18.  If  beans  are  worth  $1.60  a  bushel,  how  much 
are  they  worth  a  quart? 

19.  When  apples  sell  at  20  cents  a  peck,  what  are 
they  worth  a  bushel? 

20.  A  boy  bought  half  a  bushel  of  chestnuts  for  $1, 
and  sold  them  at  8  cents  a  quart:  how  much  did  he 
make  ? 

WRITTEN   EXERCISES. 

21.  How  many  pecks  in  12  bushels?  How  many 
quarts?     How  many  pints? 

22.  Eeduce  12  bu.  3  pk.  1  pt.  to  pints. 


1st; 

PROCESS. 

2d  PROCESS. 

bu.      pk. 

qt.      pt 

bu.       pk.    qt.     pt. 

12-^3  +  0  +  1. 

12  +  3+0  +  1. 

4 

4 

4  8,  pk. 

51,  pk. 

3- 

8 

51,  pk. 

4  0  8 ,  qt. 

8 

2 

408,  qt. 

817,  pt.,  Ans. 

2 

816,  pt. 

1 

817,  pt. 

Ans. 

DENOMINATE  NUMBERS.  147 

23.  Reduce  5  bu.  2  pk.  7  qt.  to  pints. 

24.  Eeduce  15  bu.  5  qt.  1  pt.  to  pints. 

25.  Keduce  8  bu.  3  pk.  to  quarts. 

26.  How  many  pints  in  3  pk.  5  qt.  1  pt.  ? 

27.  How  many  quarts  in  3  pk.  7  qt.  ? 

28.  How  many  pints  in  1  bu.  1   qt.  ? 

29.  How  many  bushels  in  768  pints?    In  817  pints? 

PROCESS.  PROCESS. 

2)768,  pt.  2)8J.7,  pt. 

8)384,  qt.  8)408,  qt.  +  1  pt. 

4)«^,pk.  4)51,  pk. 

12,bu.  12,  bu.  +  3  pk. 

Ans.  12  bu.  Ans.  1  2  bu.  3  pk.  1  pt. 

30.  Reduce  168  qt.  to  bushels. 

31.  Reduce  342  pt.  to  bushels. 

32.  Reduce  51  pt.  to  pecks. 

33.  How  many  pecks  in  37  pints? 

34.  How  many  bushels  in  151  quarts? 

35.  What  will  3  pk.  5  qt.  of  cherries  cost  at  5 
cents  a  pint? 

36.  A  man  sold  1  bu.  3  pk.  5  qt.  of  clover-seed 
at  8  cents  a  quart:   what  did  he  receive? 

37.  A  fruit  dealer  paid  $7  for  3  bu.  3  pk.  of  peaches, 
and  sold  them  at  75  cents  a  peck:  what  was  his  gain? 

38.  How  many  bushels  of  chestnuts  can  be  bought 
for  $15.50,  at  5  cents  a  quart? 

39.  A  fruit  dealer  put  3  bu.  2  pk.  of  strawberries 
into  quart  baskets:   how  many  baskets  were  filled? 

40.  A  boy  bought  5  pecks  of  cherries  at  60  cents 
a  peck,  and  sold  them  at  10  cents  a  quart:  how 
much  did  he  gain? 


148 


INTERMEDIATE  ARITHMETIC. 


LESSON    II. 


Art.  109.    Liquid  Measure    is   used   in    measuring 
liquids;    as,  oil,  milk,  alcohol,  etc. 

The  denominations  are  gills,    pints,   quarts,   and 
gallons. 


4  gills  {gl)  .  , 
2  pints  .  .  . 
4  quarts     .     .     . 

1  gal.  =  4  qt, 


Table. 

are  1  pint  .  .  . 
are  1  quart  .  .  . 
are  1  gallon      .     , 

=  8  pt.  =  32  gi. 


pt. 

qt 
gal 


Notes. — 1.  The  standard  liquid  gallon  contains  281  cubic  inches. 
2.  The  size  of  casks  for  liquids  is  variable.    The  capacity  of  vats, 
cisterns,  etc.,  is  usually  measured  in  barrels  of  31 J  gallons. 

1.  How    many    gills    in    3    pints?      In    10    pints? 
In  20  pints?     In  32  pints? 


DENOMINATE  NUMBERS.  149 

2.  How  many  pints   in  5  quarts?      In  81  quarts? 
In  12  quarts?     In  lOi  quarts? 

3.  How  many  quarts  in  5  gallons  ?    In  7|  gallons ? 
In  11  gallons? 

4.  How   many  pints    in    16    gills?      In    24   gills? 
In  32  gills?     In  36  gills?     In  40  gills? 

5.  How  many  quarts  in   12  pints?      In   16  pints? 
In  22  pints? 

6.  How  many  gallons  in  20  quarts?      32  quarts? 
28  quarts?     36  quarts?     40  quarts? 

7.  How  many  quarts   in   8   pints?     15   pints?     19 
pints?     13  pints?     21  pints? 

8.  How  many  pints  in  8  quarts?     11  quarts?     16 
quarts?     15^  quarts?     20  quarts? 

9.  How  many  gallons   in   8   quarts?      13   quarts? 
21  quarts?     24  quarts?     29  quarts? 

10.  How  many  quarts   in  6  gallons?      9|  gallons? 
11  gallons? 

11.  What  part  of  a  gallon   is  1   quart?     2  quarts? 
3  quarts?    4  quarts? 

12.  How  many  quarts  in  |  of  a  gallon  ? 

13.  What  will   10  quarts  of  milk  cost   at   5^   cents 
a  pint? 

14.  If  a  gallon   of  wine  cost  $6,  what  will   1  pint 
cost? 

15.  If  maple   syrup  cost  $1.60  a   gallon,  what  will 
1  quart  cost? 

16.  At  4  cents  a  pint,  what  will  5  gallons  of  milk 
cost? 

WRITTEN    EXERCISES. 

17.  How  many  pints  in  21  gallons? 

18.  How  many  gills  in  7  gal.  3  qt.  1  gi.? 

19.  How  many  pints  in  34  gal.  1  pt.  ? 


150  INTERMEDIATE   ARITHMETIC. 

20.  Ecdiice  9  gal.  2  qt.  1  j)t  to  pints. 

21.  Eediice  38  pints  to  gallons. 

22.  Eeduce  245  gills  to  gallons. 

23.  Eeduee  130  gills  to  quarts. 

24.  Eeduce  547  gills  to  gallons. 

25.  Eeduce  45|-  gallons  to  gills. 

26.  Eeduce  56  gal.  1  pt.  to  pints. 

27.  Eeduce  305  pints  to  gallons. 

28.  What  will  256  pints  of  maple  syrup  cost  at 
$1.30  a  gallon? 

29.  How  many  vials,  holding  2  gills  each,  can  be 
filled  from  a  gallon  of  alcohol  ? 

30.  How  many  jugs,  each  containing  1  gal.  2  qt., 
can  be  filled  from  a  barrel  of  vinegar  containing 
31i-  gallons? 

31.  A  grocer  bought  25  gallons  of  maple  syrup  at 
S1.20  a  gallon,  and  sold  it  at  40  cents  a  quart:  how 
much  did  he  gain? 

32.  A  grocer  bought  6  barrels  of  vinegar,  contain- 
ing 31i  gallons  each,  at  $6.50  a  barrel,  and  sold  it 
at  10  cents  a  quart:    how  much  did  he  make? 

33.  A  merchant  bought  a  hogshead  of  molasses, 
containing  63  gallons,  and  sold  |  of  it  at  75  cents 
a  gallon,  and  the  rest  at  20  cents  a  quart:  what 
did  he  receive  for  it? 

34.  A  man  bought  5  hogsheads  of  molasses,  each 
containing  63  gallons,  at  $31.50  a  hogshead,  and  sold 
3  hogsheads  at  70  cents  a  gallon,  and  2  hogsheads  at 
65  cents  a  gallon  :    how  much  did  he  gain  ? 

To  Teachers. — See  Manual  of  Arithmetic  for  addi- 
tional problems  in  Denominate  Numbers. 


DENOMINATE  NUMBERS. 


151 


LESSON    III. 


ZOJSTG    MBASUHB. 


Art.  110.  Long  Measure  is  used  in  measuring  lines 
or  distances.     It  is  also  called  Linear  Measure. 

The  denominations  are  inches,  feet,  yards,  rods, 
furlongs,  and  miles. 


Table. 


12     inches  {in.) 

.     are  1  foot     .     .     . 

.  .  ft- 

3     feet    .     .     . 

,     are  1  yard   .     .     , 

,    .    yd. 

51  yards     .     . 

.     are  1  rod     .     .     . 

.     rd. 

40    rods  .     .     . 

.     are  1  furlong   .     . 

.    fur. 

8    furlongs     . 

.    are  1  mile  .     .     . 

.     mi. 

1  mi.  =  8  fur. -320  rd.r=1760  yd. -5280  ft. -63360  in. 


152  INTERMEDIATE   AKITHMETIC. 

The  following  denominations  are  also  used : 
4  inches  are  1  hand,     [^^^%^'  measuring  the  height  of 
3  feet      are  1  pace. 

6  feet      are  1  fathom,  {  ^^^f^j.''^  measuring  the  depth  of 
3  miles    are  1  league,  j  ^^"^   '"   measuring  distances  at 

60  geographic  miles,  or,       ] 

Vare  1  degree  at  the  equator. 
69^  statute  miles  (nearly),  J 

360  degrees  ( ° )  make  the  circumference  of  the  earth. 

4  yards?      In  9  yards? 

4.  How  many  yards   in  15  feet?  In   21   feet? 

5.  How  man}^  yards  in  2  rods?  In  6  rods? 
fi.  How  many  yards  in  5  rods?  In  9  rods? 

7.  How  many  rods   in   2  furlongs?     In  5  furlongs? 
In   8  furlongs? 

8;  How  many  furlongs  in   80  rods?     In   120  rods? 

9.  How  many  furlongs  in   6  miles?     In  9  miles? 
10.  How  many   miles   in   32   furlongs?     In    56   fur- 
longs?    In  72  furlongs? 


DENOMINATE  NUMBERS.  153 

11.  How  many  rods  in  66  paces? 

12.  A  ditch  is  28  furlongs  long :  how  many  miles 
long  is  it? 

13.  A  vessel  sank  in  water  9  fathoms  deep:  what 
was  the  depth  of  water  in  feet? 

14.  A  steamer  sails  3  leagues  an  hour :  how  many 
hours  will  it  take  it  to  sail   90  miles? 

15.  A  horse  is  15  hands  high:  what  is  its  height 
in  feet? 

16.  How  many  feet  in  a  rod? 

17.  How  many  rods  in  a  mile? 

18.  What  part  of  a  foot  is  9  inches? 

19.  What  part  of  a  yard  is  2  feet? 

20.  What  part  of  a  mile  is  5  furlongs? 

WKITTEN  EXERCISES. 

21.  How  many  feet  in  16  yards?  How  many 
inches  ? 

22.  How  many  inches  in  3  fur.  20  rd.   3  j^d.? 

23.  How  many  feet  in   2  fur.  30  rd.  4ft.? 

24.  Reduce  3  mi.  5  fur.   20  rd.  to  yards. 

25.  Eeduce  1650  rods  to  miles. 

26.  Reduce  32274  inches  to  higher  denominations. 

27.  Reduce  4  mi.   27  rd.  2  ft.   10  in.  to   inches. 
'28.  How  many  steps  of  2  ft.  6  in.  each  will  a  man 

take  in  walking  2  miles? 

29.  How  many  times  will  a  wheel  6  feet  in  cir- 
cumference  turn    round    in   going    2^  miles? 

30.  Sound  travels  1090  feet  a  second:  how  many 
miles  will  it  travel  in   60  seconds? 

31.  How  many  rods  of  fence  will  be  required  to 
inclose  a  farm  which  is  -^  of  a  mile  long  and  ^  of 
a  mile  wide? 


154 


INTERMEDIATE  ARITHMETIC. 


LESSON  IV. 


Z^J\r:D   Oil  SQUAOIB  M£:ASU'EB. 


Art.  111.  Land  or 
Square  Measure  is 

used  ill  measuring 
surfaces.  It  is  also 
called  Superficial 
Measure. 

The  denomina- 
tions are  square 
inches,  square 
fee  t,  square 
yards,  square 
rods  or  perches, 
roods,  acres,  and 
square    miles. 


"iOUAR^ 


Art.   112.    A    Square    Inch 

is     a     square,     each     side     of 
which    is    an    inch    in   length. 

The  figure  at  the  left  represents 
a  square  inch  of  real  size. 


A  Square  Yard  is  a  square, 
each  side  of  which  is  a  yard, 
or  three  feet,  in  length.  It 
contains   9   square    feet. 


Note. — The  teacher  should  explain 
and  define  a  right  angle,  a  square,  a 
rectangle,  etc. 


J  rrr. 


DENOMINATE   NUMBERS.  155 

Table. 


144    square  inches  (sq. 

9     square  feet 

30 1  square  yards 

40     perches  .     . 

4    roods      .     . 

640    acres       .    . 


in.)  are  1  square  foot  .  .  .  sq.ft. 
.  are  1  square  yard  .  .  .  sq.  yd. 
.     are  1  square  rod  or  perch  P. 

.    are  1  rood E. 

.    are  1  acre A. 

.     are  1  square  mile  ,     ,     ,    sq.  mi. 

Notes. — 1.  Land  Surveyors  use  Gunter's  Chain,  which  is  4  rods 
or  66  feet  long,  and  consists  of  100  links,  each  link  being  Ty^^^^ 
inches  long.  A  square  chain  is  16  square  rods,  and  10  square 
chains  are  1  acre. 

2.  Glazing  and  stone-cutting  are  estimated  by  the  square  foot ; 
painting,  plastering,  paper-hanging,  ceiling,  and  paving,  by  the 
square  yard  ;  and  flooring,  roofing,  tiling,  and  brick-laying,  by  the 
square  of  100  feet.  Brick-laying  is  also  estimated  by  the  square 
yard,  and  by  the  1000  bricks.  For  directions  for  measuring  lum- 
ber, see  Art.  157,  Rule  6. 

1.  How  many  square  feet  in  5  square  yards?  In 
7   square  yards? 

2.  How  many  square  yards  in  36  square  feet?  In 
72   square   feet?     In   90   square   feet? 

3.  How  many  perches  in  2  roods?     In  5  roods? 

4.  How  many  roods  in  80  perches?  In  120 
perches  ? 

5.  How  many  roods   in    8   acres?     In    12   acres? 

6.  How  many  acres    in    16   roods?     In   40   roods? 

7.  How  many  square  chains  in  32  square  rods? 
In   64   square   rods?     In   80   square   rods? 

8.  How  many  acres  in  20  square  chains?  In  40 
square   chains?     In    80   square   chains? 

9.  How  many  square  yards  in  a  pavement  10  yards 
long  and  4  yards  wide? 

Solution. — In  a  pavement  10  yards  long  and  1  yard  wide 
there  are  10  square  yards,  and  in  a  pavement  10  yards  long 
and  4  yards  wide  there  are  4  times  10  square  yards,  which  is 
40  square  yards.     There  are  40  square  yards  in  the  pavement. 


150  INTERMEDIATE  ARITHMETIC. 

10.  How  many  square   yards  in   a   ceiling   8  yards 
long  and  6  yards  wide  ? 

11.  How  many  square  feet  in  a  board  16  feet  long 
and  1^  feet  wide? 

12.  How  many  perches  in  a  field  30  rods  long  and 
10  rods  wide?     How  many  roods? 

13.  How  many  square  inches   in   a  piece  of  tin  15 
inches   long   and   4  inches  wide? 

14.  How  many  square  yards  in  a  floor  15  feet  long 
and    12   feet  wide? 


^WRITTEN   EXERCISES. 

15.  How  many  square  yards  in  16  perches?  How 
manj^  square  inches? 

16.  How  many  perches  in  5  A.  2  E.  ? 

17.  Eeduce  1  A.  2  R  20  P.  10  sq.  yd.  7  sq.  ft.  to 
square  feet. 

18.  Eeduce  70882  sq.  fb.  to  higher  denominations. 

19.  Eeduce  5280  perches  to  higher  denominations. 

20.  Eeduce  5184  square  inches  to  square  yards. 

21.  How  many  acres  in  a  field  56  rods  long  and 
40  rods  wide? 

22.  How  many  acres  in  a  street  21  miles  long  and 
4  rods  wide? 

23.  How  many  square  yards  in  a  ceiling  72  feet 
long   and   40^    feet  wide? 

24.  What  will  it  cost  to  pave  a  walk  60  feet  long 
and  15  feet  wide,  at  $1.25  a  square  yard? 

25.  How  many  trees  can  be  planted  on  3  acres  of 
ground,  if  a  tree  be  planted  on  each  square  rod? 

26.  If  1000  shingles  will  cover  100  square  feet, 
how  many  shingles  will  cover  a  roof  40  feet  long 
and   25   feet  wide? 


DENOMINATE  NUMBERS. 


157 


27.  How  many  acres  of  land  in  a  township  5  miles 
square  ? 

28.  How  many  acres  in  a  township  7  miles  long 
and    6   miles  wide? 

29.  How  many  yards  of  carpeting,  a  yard  wide, 
will  carpet  a  room  20|  feet  long  and  18  feet  wide? 
/'HO.  How  many  bricks,  8  in.  long  and  4  in.  wide, 
will  pave  a  walk  60  feet  long  and   12|  feet  wide? 

^31.  What  will  it  cost  to  plaster  the  walls  and  ceil- 
ing of  a  room  15  feet  long,  12  feet  wide,  and  9  feet 
high,  at  50  cents  a  square  yard? 


LESSON    V. 


?»'^Nl50v  . 


Art.  113.  Cubic  Measure  is  used  in  measuring 
solids.     It  is  also   called   Solid  Measure. 

The  denominations  are  cubic  inches,  cubic  feet, 
and   cubic  yards. 

Art.  114.  A  cubic  inch  is  a  cube  whose  edges  are 
each  one  inch  long.  A  cubic  yard  is  a  cube  whose 
edges  are  each  one  yard  long. 

Note.— The  teacher  should  exx^lain  and  define  a  cube ;  also  its 
faces  and  edges. 


158  INTERMEDIATE   ARITHMETIC. 

Table. 

1728    cubic  inches  (cu.  in.)  are   1  cubic   foot    .     .     cu.ft. 
27    cubic  feet    ....     are   1   cubic  yard  .     .    cu.  ijd. 
1  cu.  yd.  =  27  cu.  ft.  ^  46656  cu.  in. 

Note, — A  cubic  yard  of  earth  is  called  a  load,  and  24|  cubic  feet 
of  stone  or  of  masonry  make  a  perch. 


WKITTEN  EXEKCISES. 

1.  How  many   cubic    inches   in    5   cubic  feet?      In 
12  cubic  feet?     32  cubic  feet? 

2.  How  many  cubic  feet  in  15552  cubic  inches? 

3.  How  many  cubic  feet  in  120  cubic  yards? 

4.  How  many  cubic  yards  in  405  cubic  feet? 

5.  Reduce  15  cu.  yd.  16  cu.  ft.   and  1305  cu.  in.  to 
cubic  inches. 

6.  Eeduce  1473462   cubic   inches   to   higher   denom- 
inations. 

7.  How  many  cubic  feet  in   a  block  of  marble   15 
feet  long,  12  feet  wide,  and  5  feet  thick? 


A  block  15  ft.  long,  1  ft.  wide,  and  1  ft. 
thick  contains  15  cu.  ft.;  a  block  15  ft. 
long,  12  ft.  wide,  and  1  ft.  thick  contains 
12  times  15  cu.  ft.,  or  180  cu.  ft.  A  block 
15  ft.  long,  12  ft.  wide,  and  5  ft.  thick  con- 
tains 5  times  180  cu.  ft.,  or  900  cu.  ft. 


PROCESS. 

15   cu. 

ft. 

12 

180    cu. 

ft. 

5 

900    cu. 

ft. 

8.  How  many  cubic   feet  in  a  rock   18  feet  long. 
13  feet  wide,  and  8  feet  high? 

9.  How   many   cubic    feet   in    a   pile   of  wood   24 
feet  long,  3  feet  wide,  and  8  feet  high  ? 

10.  How  many  cubic  yards   in  a  bin   9i  feet  long, 
6  feet  wide,  and  4|  feet  deep  ? 

11.  How    many    cubic    feet   of   earth    must   be   re- 


DENOMINATE  NUMBERS. 


159 


moved  to  make  a  cellar  44  feet  long,  27  feet  wide, 
and  5  feet  deep?     How  many  cubic  yards? 

.12.  How  many  cubic  yards  of  earth  must  be  re- 
moved to  make  a  reservoir  120  feet  long,  54  feet 
wide,  and  9  feet  deep  below  the  surface? 

-'  13.  What  will  it  cost  to  dig  a  cellar  36  feet  long, 
18|  feet  wide,  and  ^\  feet  deep,  at  $2.50  a  cubic 
yard? 

LESSON    VI. 


WOO  ID  mjs;a.su'R£:. 


Art.  115.  Wood  Measure  is  used  in  measuring 
wood  and  rough  stone. 

The  denominations  are  cubic  feety  cord  feet,  and 
cords. 

Table. 


16    cubic  feet     . 
8    cord  feet,  or 
128    cubic  feet 

1  cd.  =  8  cd.  ft. 


are  1  cord  foot     . 

are  1  cord  .     .     . 

128  cu.  ft. 


cd.  ft. 


160  INTERMEDIATE   ARITHMETIC. 

Note. — A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high, 
contains  1  cord  ;  and  1  foot  in  length  of  such  a  pile  contains  1  cord 
foot.  See  cut  on  page  159.  Wo(k1  four  feet,  or  nearly  four  feet,  in 
length,  is  measured  by  multiplying  the  length  of  the  pile  by  the 
height  and  dividing  the  product  by  32. 


WKITTEN   EXERCISES. 

1.  How  many  cord  feet  in  a  pile  of  wood  4  feet 
long,  4  feet  wide,  and  5  feet  high? 

2.  How  many  cubic  feet  in  6  cord  feet? 

3.  How  many  cord  feet  in  5^^  cords  of  wood? 

4.  How  many  cords  of  wood  in   128  cord  feet? 

5.  How  many  cords  of  wood  in  a  pile  containing 
1536  cubic  feet? 

6.  How  many  cords  of  wood  in  a  pile  20  feet  long, 
4  feet  w^ide,  and  6  feet  high  ? 

7.  How  many  cords  of  wood  in  a  pile  48  feet  long, 
21  feet  wide,  and   5^  feet  high  ? 

8.  A  man  bought  a  pile  of  wood  36  feet  long,  4 
feet  wide,  and  8  feet  high,  and  paid  $5.50  a  cord. 
What  did   the   wood   cost  him? 

/  9.  How  many  cords  of  stone  in  a  wall  40  rods  long, 
2  feet  thick,  and  4  feet  high  ? 

10.  At  $4.50  a  cord,  what  is  the  value  of  a  pile  of 
wood  40  feet  long,  3|  feet  wide,  and  6^  feet  high  ? 


LESSON    VII. 

Art.  116.  Circular  Measure  is  used  in  measuring 
arcs  of  circles,  and  angles,  and  in  estimating  latitude 
and  longitude.     It  is  also  called  Angular  Measure. 

The  denominations  are  seconds,  minutes,  degrees, 
signs,  and  circumferences. 


DENOMINATE  NUMBERS.  161 


Table. 

60    seconds  (^^)  .     are  1  minute     ....'' 

60    minutes  .     .     are  1  degree      .     .     .     .     ° 

30    degrees    .     .     are  1  sign S. 

12    signs,  or   ]  i     •  ^  n  - 

^^^     ,  >  .     are  1  circumference  .     .     (7.  or  cir. 

360    degrees     j 

1  cir.  -  12  S.  -  360  °  =  21600^  =  129600^^. 

Notes. — 1.  Circular  Measure  is  used  by  surveyors  in  surveying 
land;  by  navigators  in  determining  latitude 
and  longitude  at  sea;  and  by  astronomers  in 
measuring  the  motion  of  tbe  lieavenly  bodies, 
and  in  computing  dilference  in  time. 

2.  The  portion  of  surface  represented  by 
the  annexed  figure  is  a  circle. 

The  curved  line  which  bounds  the  circle 
is  its  circumference. 

Any  portion  of  a  circumference  is  an  arc, 

3.  One  half  of  a  circumference  is  called  a 
seiiii-clrcamference. 

One  fourth  of  a  circumference  is  called  a 

e  quadrant. 

One  third  of  a  quadrant  is  called  a  sign. 
A  semi -circumference    contains    180°;    a 
quadrant,  90°  ;  and  a  sign,  30°. 
4.  Every  circumference  is  divided  into  360 
equal  parts,  called  degrees,  and,  hence,  the 
length  of  a  degree  depends  upon  the  size  of 
the  circle.     A  degree  of  the  earth's  surface 
at  the  equator  contains  69^  statute  miles,  or 
CO  geographical   miles—  a  minute  of    space 
being  a  geographical  or  nautical  mile. 

1.  How  many  minutes  in  5  degrees? 

2.  How  many  signs  in  3  quadrants? 

3.  How  many  degrees  in  i  of  a  quadrant? 

4.  How  many  degrees  in  3|  signs? 

5.  How  many  signs  in  ^  of  a  cireumference  ? 


WKITTEN  EXERCISES. 

6.  How  many  seconds  in  15°  30'? 

7.  Reduce  15°  33'  to  minutes. 

T.  A.— 11. 


162 


INTERMEDIATE   ARITHMETIC. 


8.  Eeduce  5^  signs  to  minutes. 


9.  Eeduce  10800''  to  degrees. 
10.  The   sun  aj^pears   to   revolve   around   the  earth 
once   a    day :    how   many  degrees  does    it   appear  to 
pass   over   in   an    hour?     In   6   hours? 


LESSON    VIII. 
TIMB  MBA.SU'RB. 


Art.  117.  Ti  m  e 
Measure  is  used  in 
measuring  time  or 
duration. 

The  denominations 
are  seconds,  jnin- 
iites,  hours,  days, 
years,  and  centu- 
ries. 


Table. 

60    seconds  (sec)    are  1  minute   .     .     .     .  mm. 

60    minutes    .     .     are  1  hour h. 

24    hours  .     .     .     are  1  day d. 

365  days     .     .     .     are  1  common  year .     .  c.  yr. 

366  days     .     .     .     are  1  leap  year    .     .     .  I.  yr. 
100    years  (365]  d.)  are  1  century  ....  0. 

1  d  =-  24  h.  ^  1440  min.  ^  86400  sec. 

The  following  denominations  are  also  used: 

7    days      ....     are  1  week     .     .    .     .  w. 

4    weeks    ....     are  1  lunar  month      .  Ir.  m. 

13    Ir.  m.  1  d.  6  hr.,  )  ,    ^   ,.  r 

^^^,    ,  \  are  1  Julian  year  .     .  J.  yr. 

or  365}  days        j 

12    calendar  months    are  1  civil  year      .     .  e,  yr. 


DENOMINATE  NUMBERS. 


163 


Notes. — 1.  The  exact  length  of  a  solar  year  is  365  d.  5  h.  48 
min.  48  sec,  which  is  nearly  6  hours,  or  i  of  a  day,  longer  than  the 
common  year.  Since  the  common  year  lacks  I  of  a  day  of  the 
true  time,  an  additional  day  is  added  to  every  fourth  3^ear,  mak- 
ing lea})  year.  This  additional  day  is  given  to  February,  and 
hence  this  month  in  leap  year  contains  29  days.  The  leap  years 
are  exactly  divisible  by  4;  as,  1860,  1864,  1868,  1872,  etc. 

2.  The  names  and  order  of  the  calendar  months  and  the  num- 
ber of  days  in  each  are  as  follows: 


January,  1st  month,  31  days. 
February,  2d  "  28  or  29. 
March,  3d  *'  31  days. 
April,  4th       '*        30      " 

May,  5th       ''        31     '' 

June,  6th       "        30      " 


July,  7ih  month,  31  days. 

August,  8th  ''  31  " 
September,  9th  "  30  '' 
October,  10th  "  31  ^' 
November,  11th  "  30  '' 
December,  12th       ''        31     '* 


3.  The  following  couplet  will  assist  in  remembering  the  months 
which  have  30  days  each : 

Thirty  days  hath  September, 
April,  June,  and  November. 

4.  In  most  business  transactions  30  days  are  considered  a  month, 
and  360  days  a  year. 

5.  The  year  is  divided  into  four  seasons  of  three  months  each, 
as  follows: 


{March, 
April, 
May. 

{June, 
July, 
August. 


.  f  September, 

^^™f        October,     ' 
or  Fall,     [  November. 

{December, 
January, 
February. 


1.  How  many  seconds   in   5   minutes?     In   10  min- 
utes?    In  20  minutes? 

2.  How  many  minutes  in  4  hours?     In  8  hours? 

3.  How  many  hours  in  120  minutes?     In  240  min- 
utes?    In  300  minutes? 

4.  How  many  hours  in  3  days?     In  5  days? 

5.  How    many   days    in    48    hours?     In    72    hours? 
In  240  hours?     In  480  hours? 


164  INTERMEDIATE  APJTHMETIC. 

6.  How    many   days    in    6   weeks?     In    8    weeks? 
In  10  weeks?     In  15  weeks? 

7.  How  many  weeks  in  35  days?     In  49  days? 

8.  How  many  weeks  in  5   lunar  months?     In   12 
lunar  months? 

9.  How  many  lunar  months  in  IG  weeks?     In  32 
weeks?     44  weeks?     60  weeks? 

10.  How  many  calendar  months  in  5  years?  In  7 
years?     10  years?     12  years? 

'WRITTEN   EXERCISES. 

11.  How  many  seconds  in  15  hours? 

12.  How  many  hours  in  28800  seconds? 

13.  Reduce  5  d.  13  h.  40  min.  to  seconds. 

14.  Eeduce  31  d.  30  min.  45  sec.  to  seconds. 

15.  Reduce  30600  minutes  to  higher  denominations. 

16.  Reduce  52560  hours  to  common  years. 

17.  How  many  minutes  in  a  leap  year? 

18.  How  many  seconds  in  the  solar  year,  which 
contains  365  d.  5  h.  48  min.  48  sec? 

19.  How  many  seconds  in  a  common  year? 

20.  The  age  of  a  certain  man  is  64  yr.  45  d.  12  h.: 
how  many  hours  has  he  lived,  allowing  365^  days  to 
the  year? 

21.  How  many  hours  in  the  three  Spring  months? 
In  the  three  Summer  months? 

22.  How  many  minutes  will  there  be  in  the  month 
of  February,  1880?     In  February,  1882? 

23.  If  your  pulse  beat  75  times  a  minute,  how 
many  times  will  it  beat  in  5  weeks? 

/24.  How  many  days  will  it  take  a  steamship  to 
sail  3744  miles,  if  it  sail  at  the  rate  of  12  miles  an 
hour? 


DENOMINATE   NUMBERS. 


165 


LESSON    IX. 
ArOI'R^UTOIS    WBIGIIT. 


Art.  118.  Avoirdupois  Weight  is  iLsed  in  weighing 
all  articles  except  gold,  silver,  and  the  precious  stones. 

The  denominations  are  drams,  ounces,  pounds, 
hundred-weights,  and  tons. 

Table. 

16    drams  {dr.)  .     .     are  1  ounce     .     ,     ,     .     oz.    ^ 
16    ounces      .     .     .     are  1  pound    .     ...     lb. 
100    pounds     .     .     .    are  1  hundred-weight .     civt 

20    hundred-weights   are  1  ton T. 

1  T.  ^  20  cwt.  =  2000  Ih.  =  32000  oz.  -=  512000  dr. 

196    pounds  of  flour,       )  ,  ,         , 

^^^    „  ,  ,      ^       >      .     .     .     .     are  1  barrel.    ^ 

200    lb.  pork  or  beef,      j 

100  lb.  of  fish are  1  quintal. 

14  lb.  of  lead  or  iron are  1  stone. 

56  lb.  of  corn,  rye,  or  flax-seed, ") 

60  lb.  of  wheat  or  clover-seed,    V  .     are  1  bushel. 

32  lb.  of  oats,  j 


166  INTERMEDIATE  ARITHMETIC. 

Notes. — 1.  In  wholesaling  and  freighting  coal  and  in  invoicing 
Englisli  goods  at  the  United  States  custom-houses,  the  hundred- 
weight is  divided  into  4  quarters,  of  28  pounds  each,  and  tlie  ton 
contains  2240  pounds.  This  is  called  the  long  or  gross  ton,  while 
the  ton  of  2000  pounds  is  called  the  short  or  net  ton. 

2.  The  dram  is  seldom  used  in  business  transactions,  and  the 
quarter,  of  25  pounds,  is  never  used. 


1.  How  many  drams  in  2  ounces?     In  5|^  ounces? 
10  ounces?     15  ounces? 

2.  How  many  ounces  in  48  drams?     In  64  drams? 
96  drams?     160  drams? 

3.  How  many  ounces  in  4  pounds?     In  6|  pounds? 
lOf  pounds?     12^  pounds? 

4.  How  many  pounds  in  80  ounces? 

5.  How  many  pounds  in  5  hundred-weight?     In  8 
cwt?     12f  cwt.?     25  cwt? 

6.  How    many    hundred-weight    in    4    tons?     In    6 
tons?     8f  tons?     12|  tons? 

7.  What  will  |  of  a  pound  of  candy  cost  at  2  cents 
an  ounce? 

8.  What  will  |  of  a  hundred-weight  of  flour  cost  at 
5  cents  a  pound? 

WRITTEN   EXERCISES. 

9.  Reduce  5  tons  to  ounces. 

10.  Reduce  3  T.  14  cwt.  56  lb.  to  pounds. 

11.  Reduce  5  cwt.  77  \h.  13  oz.  to  ounces. 

12.  Reduce  34920  pounds  to  tons. 

13.  Reduce  4560  ounces  to  higher  denominations. 

14.  Reduce  11  T.  38  lb.  15  oz.  to  drams. 

15.  What  will   a   barrel   of  flour   cost   at   6   cents   a 
pound  ? 

16.  What  will   3  barrels  of  pork  cost  at  15  cents  a 
pound? 


DENOMINATE   NUMBERS. 


167 


17.  How  many  barrels  will  3920  pounds  of  flour 
make  ? 

18.  A  farmer  sold  3600  pounds  of  wheat  at  $1.75  a 
bushel :    how  much  did  he  receive  ? 

19.  A  hay-stack  contains  9000  pounds  of  hay:  what 
is  it  worth  at  $12  a  ton? 

^  20.  What  will    it   cost   to   transport   50   T.  15  cwt. 
75  lb.  of  freight,  at  ^  cent  a  pound  ? 

21.  A  farmer  exchanged  45|  pounds  of  butter,  at 
20  cents  a  pound,  for  sugar,  at  15  cents  a  pound: 
how  much  sugar  did  he  receive? 


LESSON    X. 


Art.  119.   Troy  Weight 

is  used  in  weighing  gold, 
silver,  and  precious  stones, 
and  also  in  philosophical 
exj)eriments. 

The  denominations  are 
grains,  pennyweights, 
ounces,  and   -pounds. 


Table. 


24    grains  (^r.)     .     are  1  pennyweight  .     .  pwt. 

20    pennyweights     are  1  ounce     ....  02. 

12    ounces   .     .     .     are  1  pound    .     .     .     .  ?6. 

1  lb.  =  12  oz.  =  240  pwt.  =--  5760  gr. 


168  INTERxMEDIATE   ARITHMETIC. 

Notes. — 1.  Diamonds  are  weighed  by  carats  and  fractions  of 
carats.     A  carat  is  4  Troy  grains. 

2.  The  purity  of  gold  is  also  expressed  in  carats,  a  carat  mean- 
ing ^1^  part.  Gold  that  is  22  carats  fine  contains  22  parts  of  pure 
gold,  and  2  parts  o^  alloy. 

1.  How    many   grains   in    5   penny weigbtB?     In   3 
pwt.?     8  pwt.?     10^  pwt.? 

2.  How  many  pennyweights   in    3   ounces?     In    6 
ounces?     9  ounces?     10  ounces? 

3.  How  many  ounces  in  40  pennyweights?     In  80 
pwt.?     100  pwt.?     120  pwt.? 

4.  How  many  ounces  in  4  pounds?    In  7^  pounds? 
12|  pounds?     20  pounds? 

5.  How  manj^  pounds  in  36  ounces?    In  60  ounces? 
84  ounces?     96  ounces? 

6.  What   part  of  a  pound  is  1  ounce?     6  ounces? 
8  ounces?     9  ounces? 


-WRITTEN  EXERCISES. 

7.  Eeduce  44  lb.  3  oz.  13  pwt.  to  pennyweights. 

8.  Eeduce  7  oz.  15  pwt.  to  grains. 

9.  Eeduce  56  lb.  13  pwt.  to  grains. 

10.  Eeduce  13486  pwt.  to  higher  denominations. 

11.  Eeduce  40408  grains  to  higher  denominations. 

12.  Eeduce  5680  ounces  to  pounds. 

13.  Eeduce  5280  grains  to  ounces. 

14.  A  lady  bought  a  pearl  necklace,  weighing  8  oz. 
15  pwt,  at  75  cents  a  grain:   what  did  it  cost? 

15.  What  will  be  the  cost  of  a  gold  chain,  weighing 
31  oz.,  at  75  cents  a  pwt.? 

46.  How  many  gold    rings,   each  weighing   3  pwt., 
can   be    made   from   a   bar   of  gold   weighing  |   of  a 
? 


pound  ? 


DENOMINATE  NUMBERS. 


169 


LESSON    XI. 


Art.  120.  Apothecaries 
Weight  is  used  by  physi- 
cians in  prescribing  and 
by  apothecaries  in  mixing 
medicines. 

I        The    denominations    are 

grains,  scruples,  drams,     .-^^ 
ounces,  and  pounds. 


Table. 


20 

grains  {gr.) 

.     are   1   scruple 

.    .  9. 

3 

scruples 

.     are   1   dram      .     . 

•    .    3- 

8 

drams    .     . 

are   1   ounce     .     . 

.    .    I. 

12 

ounces  .     . 

are    1   pound    .     . 

.    .   lb. 

Note.  —  Medicines  are  bought  and  sold  in  quantities  by  avoir- 
dupois weight. 

1.  How  many  grains  in  2  scruples? 

2.  How  many  scruples  in  5  drams?     In  7  drams? 

9  drams?     12  drams?     20  drams? 

3.  How  many  drams  in  21  scruples?     In  27  scru- 
ples?    33  scruples?     40  scruples? 

4.  How  many  drams  in  5  ounces?     In  8  ounces? 

10  ounces?     12  ounces? 

5.  How  many  pounds  in  36  ounces?    In  72  ounces? 
96  ounces?     120  ounces? 

6.  How  many  ounces  in  5  pounds?     In  8  pounds? 
10^  pounds?     12  pounds? 


170 


INTERMEDIATE  ARITHMETIC. 


\?VrRITTEN   EXERCISES. 

7.  Eeduce  16  lb.  11  g  5  5  2  9  10  gr.  to  grains. 

8.  Keduce  10  g  83  to  grains. 

9.  Keduce  356  5  to  pounds. 

10.  Eeduce  26484  gr.  to  higher  denominations. 

11.  How  many  pounds  in  5760  9? 

12.  How  many  doses,  of  18  gr.  each,  in  5  3  2  9  of 
tartar  emetic? 

13.  How  many  pills,   of  5   gr.    each,   can   be   made 
from  1  §  2  3  2  9  of  calomel  ? 

14.  How   many  ounces   of  calomel    will    make   480 
pills,  each  weighing  6  grains? 


LESSON    XII. 
MISCBZLAJVJBJOUS    T;A^LB. 

PAPER. 

24  sheets      are  1  quire. 

20   quires      are  1  ream. 

2   reams      are  1  bundle. 

5  bundles  are  1  bale. 


12   things  are   1   dozen. 

12   dozen    are   1   gross. 

1 2  gross     are   1   great  gross. 

20    things  are   1   score. 

Note.— A  sheet  of  paper  folded  in  2  leaves  is  called  a  folio;  in 
4  leaves,  a  quarto,  or  4to;  in  8  leaves,  an  octavo,  or  8vo;  in  12  leaves, 
a  duodecimo,  or  12mo ;  in  18  leaves,  an  ISmo. 

1.  How  many  sheets  of  paper  in  5|-  quires? 

2.  How  many  quires  of  paper  in  4  reams?     In  8 
reams?     12|  reams?     15  reams? 


DENOMINATE  NUMBERS.  171 

3.  How  many  bundles   of  paper  in   6  reams?     In 
12  reams?     18  reams?     32  reams? 

4.  How   many   eggs   in    5   dozen?     In    7f   dozen? 
8^-  dozen?     12  dozen?     20  dozen? 

5.  How  many  years  are  4  score   years?     3  scoro 
years  and  10? 

WRITTEN   EXERCISES. 

6.  How  many  sheets  of  paper  in  12i  reams? 

7.  Eeduce  6  rm.  15  qu.  12  sheets  to  sheets. 

8.  What  will  7200  sheets  of  paper  cost  at  $8.50  a 
ream? 

9.  How  many  crayons   are  there   in   36   boxes,  if 
each  box  contains  1  gross? 

10.  If  a  shirt  require  6  buttons,  how  many  shirts 
will  12  gross  of  buttons  trim? 

11.  What  will  44  gross  of  lead-pencils  cost  at  75 
cents  a  dozen? 

12.  A  stationer  bought  15  reams  of  letter-paper  at 
$3.50  a  ream,  and  sold  it  at  25  cents  a  quire:  how 
much  did  he  gain? 


LESSON    XIII. 

DEFINITIONS,  PEINOIPLES,  AND  RULES. 

Art.  121.  A  Denominate  Number  is  a  number 
composed  of  concrete  units  of  one  or  several  denom- 
inations. 

Art.  122.  Denominate  Numbers  are  either  Simple 
or  Compound, 

A  Simple  Denominate  Number  is  composed  of 
units  of  the  same  denomination ;   as,  7  quarts. 


172  INTERMEDIATE  ARITHMETIC. 

A  Compound  Denominate  Num^her  is  composed 
of  units  of  several  denominations;  as,  5  bu.  3  pk. 
7  qt. 

Note.  —  Compound  Denominate  Numbers  are  properly  called 
Compound  Numbers,  since  every  compound  number  is  necessarily 
denominate. 

Art.  123.  Denominate  Numbers  express  Ciirrenc[j , 
Measure,  and   Weight. 

Carreiicy  is  the  circulating  medium  used  in  trade 
and  commerce  as  a  representative  of  value. 

Measure  is  the  representation  of  extent,  capacity, 
or  amount. 

Weight  is  a  measure  of  the  'force  called  gravity, 
by  which  bodies  are  drawn  toward  the  earth. 

Art.  124.  The  following  diagram  represents  the 
three  general  classes  of  denominate  numbers,  their 
subdivisions,  and  the  tables  included  under  each : 

(1.  Coin,  ) 

1.  Currency,  \  \  United  States  Money. 

(2.  Paper  Money,    j 

1.  Lines,     P-  ^^'^ 
or  arcs,     1  2    ci 


2.  Measure, 


1.  Of  extension, 


1 .  Long  Measure. 
Circular  Measure. 

2.  Surfaces :  Square  Measure. 

ri.  Cubic  Measure, 
o    n r.x4-,.       2.  Wood  Measure. 

3.  Capacity,      3    j^^^  Measure. 

[4.  Liquid  Measure. 


.2.  Of  duration:  Time  Measure, 
fl.  Avoirdupois  Weight. 
3.  Weight,  -j  2.  Troy  Weight. 

[3.  Apothecaries  Weight. 

Art.  125.  The  Heduction  of  a  denominate  number 


DENOMINATE  NUMBERS.  173 

is  the  process  of  changing  it  from  one  denomination 
to  anotlier  without  altering  its  value. 

Art.  126.  Eeduction  is  of  two  kinds:  Eeducfion 
Descending  and  Eeduction  Ascending, 

Reduction  Descendhtg  is  the  process  of  changing 
a  denominate  number  from  a  higher  to  a  lower  de- 
nomination.    It  is  performed  by  multij)lication. 

Reduction  Ascending  is  the  process  of  changing 
a  denominate  number  from  a  lower  to  a  higher  de- 
nomination.    It  is  performed  by  division. 

Art.    127.    EuLE     for    Eeduction    Descending. — 

1.  Midtiply  the  number  of  the  highest  denomination  by 
the  number  of  units  of  die  next  loiver  ivhich  equals  a  unit 
of  the  higher,  and  to  the  product  add  the  number  of  the 
loiver   denomination,  if  any. 

2.  Proceed  in  like  manne)*  ivith  this  and  each  successive 
result  thus  obtained  until  the  number  is  reduced  to  the  re- 
quired denomination. 

Note. — Tlie  successive  denominations  of  the  compound  num- 
ber sliould  be  written  in  their  proper  order,  and  the  vacant  de- 
nominations, if  any,  filled  with  ciphers. 

Art.    128.     EuLE     for    Eeduction    Ascending.  — 

1.  Divide  tlie  given  denominate  number  by  the  number 
of  units  of  its  oivn  denomination  which  equals  one  unit  of 
the  next  higher,  and  place  the  remainder,  if  any,  at  the 
right. 

2.  Proceed  in  like  manner  tvith  this  and  each  successive 
quotient  thus  obtained  until  the  number  is  reduced  to  the  re- 
qidred  denomination. 

3.  The  last  quotient,  with  the  several  remainders  annexed 
in  proper  orda%  will  be  the  answer  required. 


174  INTERMEDIATE    ARITHMETIC. 


Questions  for  Keview.  ' 

What  is  a  number?  What  is  an  abstract  number?  What 
is  a  denominate  number  ?  Into  what  two  classes  are  denom- 
inate numbers  divided?    Define  each. 

By  what  other  names  are  compound  denominate  numbers 
usually  called  ?  Why  may  the  word  "  denominate "  be 
omitted?  What  is  currency  ?  Of  how  many  kinds  of  money 
is  United  States  currency  composed  ? 

What  is  meant  by  measure  ?  Name  the  two  kinds  of 
measures.     How  are   the   measures   of   extension   divided? 

What  tables  are  used  in  measuring  lines?  Surfaces?-  Con- 
tents? Capacity?  What  table  is  used  in  measuring  duration? 
What  ones  are  used  in  measuring  the  weight  of  bodies? 

What  is  reduction  ?  Name  the  two  kinds  of  reduction. 
Define  each  kind.     Repeat  the  rule  for  each. 

For  what  is  Dry  Measure  used?  Name  the  denominations. 
Repeat  the  table.  For  what  is  Liquid  Measure  used  ? 
Name  the  denominations.  Repeat  the  table.  For  what  is 
Long  Measure  used?  Name  the  denominations.  Repeat 
the   table. 

For  what  is  Square  Measure  used  ?  Name  the  denomina- 
tions. Repeat  the  table.  What  is  a  square  inch  ?  A  square 
yard  ?  For  what  is  Cubic  Measure  used  ?  Name  the  denom- 
inations. Repeat  the  table.  What  is  a  cubic  inch  ?  A  cubic 
yard  ?  For  what  is  Wood  Measure  used  ?  Name  the  de- 
nominations.    Repeat  the  table. 

For  what  is  Circular  Measure  used  ?  Name  the  denomina- 
tions. Repeat  the  table.  For  what  is  Time  Measure  used? 
Name  the  denominations.     Repeat  the  table. 

Name  the  calendar  months  in  their  order,  and  give  the 
number  of  days  in  each.  How  many  days  has  February  in 
leap  years?  Name  the  four  seasons  of  the  year,  and  the 
months  of  each. 

For  what  are  the  three  weights  respectively  used?  Give 
the  denominations  and  repeat  the  table  of  each.  Repeat  the 
miscellaneous  table. 


DENOMINATE  NUMBERS.  175 

LESSON    XIV. 
Miscellaneous   ^erte}p   ^7*oblems, 

1.  How  many  quarts  in  |  of  a  bushel? 

2.  How  many  pints  in  3|  gallons? 

3.  How  many  hours  in  ^  of  a  week  ? 

4.  How  many  ounces  in  2|-  pounds  of  sugar? 

5.  What  will  -I  of  a  cwt.  of  sugar  cost  at  15  cents 
a  pound? 

6.  What  will  I  of  a  gallon  of  oil  cost  at  25  cents  a 
pint? 

7.  What  will  f  of  a  ream  of  paper  cost  at  20  cents 
a  quire? 

8.  What  costs  -|  of  a  ton  of  hay  at  75  cents  a  cwt.  ? 

9.  A  boy  picked  3  pecks  of  cherries,  and  sold  them 
at  10  cents  a  pint;  how  much  did  he  receive? 

10.  If  a  ship  sail  3  leagues  an  hour,  in  how  many 
hours  will  it  sail  63  miles? 

11.  How  many  half-pint  bottles  will  a  gallon  of 
sweet   oil   fill? 

12.  How  many  quart  baskets  will  3  pk.  5  qt.  of 
strawberries   fill  ? 

13.  How  many  leap  years  in  every  century? 

14.  How  many  calendar  months  in  20  years? 

WRITTEN  EXERCISES. 

15.  A  fruit  dealer  bought  24  barrels  of  apples,  con- 
taining 2|  bushels  each,  at  $2.50  a  barrel,  and  sold 
them  at  $1.25  a  bushel;  what  was  his  gain? 

16.  What  will  20  yd.  2  ft.  of  iron  railing  cost  at 
$1.25  a  foot? 

17.  What  will  40  miles  of  telegraph  wire  cost  at 
25   cents   a   yard? 


176  INTERMEDIATE  ARITHMETIC. 

18.  How  many  times  will  a  carriage- wheel  11  feet 
in  circumference  turn  round  in  running  2  miles? 

19.  How  many  times  will  a  car- wheel  5  feet  in  cir- 
cumference turn  round  in  running  from  Columbus  to 
Cincinnati,  the  distance  being  120  miles? 

20.  How  many  acres  in  a  township  6  miles  square? 

21.  What  will  a  piece  of  land  40  rods  long  and  32 
rods  wide  cost  at  $75  an  acre? 

22.  How  many  hills  of  corn  can  be  planted  on  5 
acres,  allowing  1  hill  to  every  square  yard? 

23.  How  many  people  can  stand  on  a  terrace  250 
feet  long  and  120  feet  wide,  allowing  4  persons  to 
each   square    yard  ? 

24.  What  will  it  cost  to  gravel  a  street  129  rods 
long  and  60  feet  wide  at  75  cents  a  square  yard? 

'  25.  How  many  square  yards  in  the  walls  and  ceiling 
of  a  room  21  ft.  long,  18  ft.  wide,  and  9  ft.  high? 

'  26.  How  man}^  yards  of  carpeting,  a  yard  wide,  will 
carpet  a  room   18^  feet  long  and  15  feet  wide? 

'  27.  If  1000  shingles  will  cover  100  square  feet,  how 
many  shingles  will  cover  a  roof  each  side  of  which  is 
48  feet  long  and  15  feet  wide? 

28.  A  park  containing  40  acres  is  50  rods  wide:  how 
long  is  it? 

29.  How  many  cubic  feet  in  a  bin  12  feet  long,  8 
feet  wide,  and  3|-  feet  deep? 

30.  How  many  perches  of  stone  in  a  wall  99  feet 
long,  8  feet  high,  and  \\  feet  thick? 

31.  What  will  it  cost  to  dig  a  ditch  80  rods  long,  \\ 
feet  wide,  and  2  feet  deep,  at  15  cents  a  cubic  yard? 

32.  At  $4.50  a  cord,  what  will  be  the  cost  of  a  pile 
of  wood  48  feet  long,  6  feet  high,  and  4  feet  wide? 

33.  Ho\v  many  times  will  a  clock  that  ticks  seconds 
tick  in  the  month  of  June? 


COMPOUND  NUMBERS.  177 

34.  If  a  person  read  a  half  hour  each  day,  how  many 
hours  will  he  read  in  40  years,  of  365^  days  each? 

35.  How  many  gold  rings,  each  weighing  4  pwt., 
can  be  made  from  a  bar  of  gold  w^eighing  1  lb.  4  oz.  ? 

36.  A  car  contains  80  barrels  of  pork,  and  another 
80  barrels  of  flour:  what  is  the  difference  in  the 
freight   of   the  tw^o   cars? 

37.  How  many  gross  of  pens  will  supply  4320  pupils 
one  year,  if  each  pupil  require  4  pens? 

38.  If  10  sheets  of  paper  will  make  a  16mo.  book  of 
320  pages,  how  many  reams  will  it  take  to  publish  an 
edition  of  2000  copies? 


SECTION  XI. 


LESSON    I. 
Addition  of  Compound  A^ambers, 

1.  What  is  the  sum  of  5  bu.  3  pk.  6  qt.  1  pt. ;  8  bu. 
2  pk.  1  pt. ;  10  bu.  1  pk.  3  qt. ;  and  3  pk.  5  qt.  1  pt.? 

Write  the  compound  numbers 
so  that  terms  of  the  same  denom- 
ination shall  stand  in  the  same 
column.  Add  first  the  column 
of  pints.  The  snm  is  3  pints, 
which  equals  1  qt.  1  pt.     Write 

25  bu.    2  pk.      7  qt.    1  pt.      the  1  pt.  under  the   pints,  and 

add  the  1  qt.  with  the  column 

of  quarts.  The  sum  of  the  quarts  is  15  quarts,  which  equals 
I.  A.  12. 


PROCESS 

5. 

bu. 

pk. 

qt. 

pt. 

5 

3 

6 

1 

8 

2 

0 

1 

10 

1 

3 

0 

8 

5 

1 

178  INTERMEDIATE  ARITHMETIC. 

1  pk.  7  qt.  Write  the  7  qt.  under  the  quarts,  and  add  the 
1  pk.  with  the  column  of  pecks.  The  sum  of  the  pecks 
is  10  pecks,  which  equals  2  bu.  2  pk.  Write  the  2  pk.  un- 
der the  pecks,  and  add  the  2  bu.  with  the  column  of  bushels. 
The  sum  of  the  bushels  is  25  bushels.  The  sum  of  the  four 
compound  numbers  added  is  25  bu.  2  pk.  7  qt.  1  pt. 

(2)  (3)  (4) 

bu.  pk.  qt.  pt.       gal.   qt.    pt.  gi.         mi.  fur.  rd.    yd,    ft.    in. 


16 

2  6 

1 

21 

3 

1   3 

19 

7   39   5 

2 

10 

23 

1   4 

0 

16 

0 

1   2 

27 

3   24   3 

1 

6 

40 

3  0 

1 

48 

2 

0  0 

45 

4   33   0 

0 

7 

9 

0   2 

0 

35 

0 

1   3 

(6) 

- 

6 

0   17   2 

(7) 

1 

0 

(5) 

cwt 

.  lb.  oz. 

dr. 

lb. 

OZ.  pvvt. 

gr. 

lb.  S.  5. 

9. 

gr. 

15 

63  11 

13 

9 

11  19 

23 

44  11  7 

2 

19 

18 

85   0 

10 

18 

6  13 

20 

23   9  6 

0 

8 

6 

15  15 

0 

7 

10   8 

11 

10  5 

2 

16 

0 

75   8 

7 

9  15 

16 

27   7  6 

1 

14 

19 

36  14 

15 

23 

0  10 

9 

16   3  0 

0 

18 

8.  What  is  the  sum  of  15  w.  5  d.  22  h.  45  min.  34 
sec. ;  8  w.  6  d.  13  h. ;  3  w.  20  h.  52  min. ;  4  d.  22  h. 
33  min.  55  sec;  1  w.  2  d.  3  h.  30  min.? 

9.  Add  14°  30'  46'^;  53°  16'  49";  26°  34'  15";  18° 
44'  33";    62°  36';    and  43°  45". 

10.  Add  5  sq.  mi.  625  A.  3  E.  35  P. ;  14  sq.  mi.  546 
A.  2  K.  28  P. ;  486  A.  1  R.  27  P. ;  94  A.  24  P. ;  and 
14  sq.  mi.  300  A.  3  R.  36  P. 

11.  A  wood  dealer  bought  5  piles  of  wood,  the  first 
containing  21  cd.  5  cd.  ft.  15  cu.  ft. ;  the  second,  45  cd, 
12  cu.  ft. ;  the  third,  18  cd.  7  cd.  ft. ;  the  fourth,  50  cd. 
6  cd.  ft.  14  cu.  ft;  and  the  fifth,  16  cd.  5  cd.  ft:  how 
much  wood  did  he  purchase? 

12.  A  printer  used  3  bundles  1  ream  16  quires  of 
paper,  Monday ;  2  bundles  1  ream,  Tuesday ;  4  bundles 


r 


COMPOUND  NUMBERS.  179 

16  quires,  Wednesday;  3  bundles  1  ream  18  quires, 
Thursday;  5  bundles,  Friday;  and  3  bundles  1  ream, 
Saturday:    how  much  paper  did  he  use? 

13.  The  four  quarters  of  an  ox  weighed  respectively 
2  cwt.  84  lb.  10  oz. ;  3  cwt.  1  lb.  14  oz. ;  2  cwt.  76  lb. 
4  oz. ;  and  2  cwt.  98  lb.  14  oz :  what  was  the  weight 
of  the  four  quarters  ? 

14.  A  garden  has  four  unequal  sides;  the  first  is  4 
rd.  3  yd.  2  ft.  8  in.;  the  second,  5  rd.  1  ft.  10  in.;  the 
third,  4  rd.  5  yd.  4  in. ;  and  the  fourth,  3  rd.  4  yd.  2  ft. 
9  in.:   what  is  the  distance  round  the  garden? 

15.  A  cistern  full  of  water  was  emptied  by  3  pipes ; 
the  first  discharged  45  gal.  3  qt. :  the  second,  54  gal. 
1  pt. ;  and  the  third,  61  gal.  2  qt,  1  pt. :  how  much 
water  did  the  cistern  contain. 


DEFINITIONS,  PEINCIPLE,  AND  RULE. 

Art.  129.  A  Cofnpoitfid  Number  is  a  number  com- 
posed of  units  of  several  denominations. 

Art.  130.  The  numbers  expressing  the  successive  de- 
nominations of  a  compound  number  are  called  its 
Tejvns. 

Compound  numbers  are  of  the  sajne  hind  when 
their  corresponding  terms  express  units  of  the  same 
denomination;    as,  3  bu.  2  pk.,  and  6  bu.  3  pk.  5  qt. 

Art.  131.  Compound  Addition  is  the  process  of 
finding  the  sum  of  two  or  more  compound  numbers 
of  the  same  kind. 

Art.  132.  Principle. — In  both  simple  and  compound 
addition,  the  sum  of  each  column  is  divided  by  the  num- 
ber of  units  of  that  denomination  which  equals  one  of  the 


180  INTERMEDIATE   ARITHMETIC. 

next  higher  denomination.  In  simple  addition  this  di- 
visor is  10;  in  compound  addition  it  is  a  varying 
number,  since  the  several  denominations  are  expressed 
on  a  varying  scale. 

Art.  133.  Rule. — 1.  Write  the  compound  numbers  to 
to  he  added  so  that  units  of  the  same  denomination  shall 
stand  in  the  same  column. 

2.  Add  first  the  column  of  the  lowest  denomination^ 
and  divide  the  sum  by  the  number  of  units  of  that  de- 
nomination which  equals  a  unit  of  the  next  higher  de- 
nomination ;  write  the  remainder  under  the  column  added, 
and  add  the  quotient  with  the  next  column. 

3.  In  like  manner  add  the  remaining  columns^  writing 
the  sum  of  the  highest  column  under  it. 


LESSON    II. 
SuhtractloH  of  Co?7zpou?id  JVumbers, 

1.  From  13  lb.  5  oz.  16  pwt.  21  gr.  take  9  lb.  4  oz. 
18  pwt.  15  gr. 

Write  the  subtrahend  under 
the  minuend,  placing  terms  of 
the  same  denomination  in  the 
same  column.  Subtract  15  gr. 
4  lb.  0  oz.  18  pwt.  6  gr.  fi'««i  21  gr.,  and  write  6  gr., 
the  difFeretice,  under  the  grains. 
Since  18  pwt.  are  greater  than  16  pwt.,  add  20  pwt.  to  16 
pwt.,  making  36  pwt.  Subtract  18  pwt.  from  36  pwt.,  and 
write  18  pwt.,  the  difference,  under  the  pennyweights.  Since 
20  pwt.  were  added  to  the  minuend,  add  1  oz.  (which  equals 
20  pwt.)  to  the  4  oz.  of  the  subtrahend,  making  5  oz.  Sub- 
tract 5  oz,  from  5  oz.,  and  write  0  oz.,  the  difference,  under 


PROCESS. 

lb. 

oz.     pwt. 

gr- 

13 

5        16 

21 

9 

4        18 

15 

COMPOUND  NUMBERS.  181 

the  ounces.  Subtract  9  lb.  from  13  lb.,  and  write  4  lb.,  the 
difference,  under  pounds.  The  difference  is  4  lb.  18  pwt. 
6  gr. 

(2)                                           (3)  (4) 

cwt.  lb.  oz.  dr.        lb.    g.    5.   9.   gr.  w.    d.    h.  min. 

From  48    73     10     15          7     10    7     1     14  13     1     13     45 

Take    29    47     14      9          3     11     5     2     16  8    6     17     33 

(5)                                        (6)  (7) 

mi.  fur.  rd.    yd.           rd.  yd.  ft.   in.  gal.  qt.    pt.  gi. 

From  405    5     25      4            35     5     2     10  44      3      1      2 

Take    384    6     37      5            27     4     1     11  26      3      1      3 


8.  A  farmer  raised  7  bu.  1  pk.  4  qt.  of  clover-seed, 
and  sold  5  bu.  6  qt.  1  pt. :    how  much  had  he  left  ? 

9.  A  man  bought  a  farm  containing  356  A.  2  R.  25 
P.,  and  sold  148  A.  3  R  36  P. :  how  much  land  had 
he  left? 

10.  Washington  is  77°  2'  48"  W.  longitude,  and 
San  Francisco  122°  26'  15"  W.  longitude:  how  much 
farther  west  is  San  Francisco  than  Washington? 

11.  From  a  stack  of  hay  containing  5^  tons,  a 
farmer  sold  3  T.  12  cwt.  65  lb. :  how  much  hay  re- 
mained unsold? 

12.  From  a  hogshead  of  molasses  containing  63 
gallons,  a  grocer  sold  38  gal.  3  qt.  1  pt. :  how  much 
molasses  remained  in  the  hogshead? 

13.  A  silversmith  bought  a  bar  of  gold  weighing 
1  lb.  5  oz.  12  pwt.,  and  a  bar  of  silver  weighing 
3  lb.  8  oz.  16  pwt.  10  gr. :  how  much  more  silver 
than  gold  did  he  buy? 

14.  A  company  contracted  to  build  65  mi.  4  fur. 
of  railroad,  and  completed  the  first  year  27  mi.  7  fur. 
20  rd. :    how  much  remained  to  be  built? 


PROCESS. 

y- 

mo. 

d. 

1869, 

9 

15 

1863, 

7 

23 

6y] 

r.   1  mo. 

,  22d. 

182  INTERMEDIATE  ARITHMETIC. 

15.  A  note  was  given  July  23,  1863,  and  paid  Sept. 
15,  1869:    how  long  did  it  run? 

Write  the  earlier  date  under  the 
later,  writing  the  number  of  the  year, 
month,  and  day  in  proper  order,  and 
subtract,  allowing  30  days  to  a  month, 
and  12  months  to  a  year. 

16.  What  is  the  difference  of  time  between  Oct.  23, 
1856,  and  June  15,  1866? 

17.  How  long  from  April  12,  1861,  to  May  22,  1865? 

18.  Abraham  Lincoln  was  born  Feb.  12,  1809,  and 
died  April  15,  1865:    what  was  his  age? 

19.  The  American  Eevolution  began  April  19,  1775, 
and  ended  Jan.  20,  1783:  how  long  did  it  continue? 

20.  America  was  discovered  Oct.  14,  1492,  and  the 
Declaration  of  Independence  was  signed  July  4,  1776 : 
how  much  time  elapsed  between  these  two  events? 

21.  The  laying  of  the  Atlantic  Cable  was  consum- 
mated July  28,  1866,  and  the  Pacific  Railroad  was 
completed  May  10,  1869:  how  much  earlier  was  the 
first  event  than  the  second? 

22.  Andrew  Jackson  died  at  Nashville,  Tenn.,  June 
8,  1845,  aged  78  yr.  2  mo.  23  days:  what  was  the 
date  of  his  birth? 


DEFIITITION  AO  EULE. 

Art.  134.  Compound  Subtraction  is  the  process 
of  finding  the  difference  between  two  compound  num- 
bers of  the  same  kind. 

Art.  135.  EuLE. — 1.  Write  the  subtrahend  under  the 
minuend,  placing  terms  of  the  same  denomination  in  the 
same  column. 


COMPOUND  NUMBERS.  183 

2.  Beginning  at  the  rights  subtract  each  successive 
term  of  the  subtrahend  from  the  corresponding  term  of 
the  minuend^  and  write  the  difference  beneath. 

3.  If  any  term  of  the  subtrahend  he  greater  than  the 
corresponding  term  of  the  minuend^  add  to  the  term  of 
the  minuend  as  many  units  of  that  denomination  as  equal 
one  of  the  next  higher^  and  from  the  sum  subtract  the 
term  of  the  subtrahend^  writing  the  difference  beneath. 

4.  Add  one  to  the  next  term  of  the  subtrahend^  and 
proceed  as  before. 

Note, — Instead  of  adding  one  to  the  next  term  of  the  subtra- 
hend, one  may  be  subtracted  from  the  next  term  of  the  minuend. 


LESSON     III. 
Multiplicatio7i  of  Compound  A^umbers. 

1.  Multiply  34  gal.  3  qt.  1  pt.  by  9. 

PROCESS.  Write   the    multiplier    under    the 

lowest  denomination  of  the  multipli- 
cand.    9  times  1  pt.  are  9  pt.,  equal 
to  4  qt.  1  pt.     Write  the  1  pt.  under 
313  gal.  3  qt.    1  pt.  pints,  and  reserve  the  4  qt.  to  add  to 

the  product  of  quarts.  9  times  3  qt. 
are  27  qt.,  and  4  qt.  added  are  31  qt.,  equal  to  7  gal.  3  qt. 
Write  the  3  qt.  under  quarts,  and  reserve  the  7  gal.  to  add 
to  the  product  of  gallons.  9  times  34  gal.  are  306  gal.,  and 
7  gal.  added  are  313  gal.  Hence,  9  times  34  gal.  3  qt.  1  pt. 
=  313  gal.  3  qt.  1  pt. 


gal. 

qt. 

pt. 

34 

3 

1 
9 

(2) 

(3) 

(4) 

lb. 

oz. 

pwt. 

yd. 

ft. 

in. 

mi. 

fur. 

rd. 

15 

6 

13 

8 

7 

2 

11 
12 

15 

3 

22 
6 

184  INTERMEDIATE   ARITHMETIC. 


(5) 

(6 

) 

(7) 

lb. 

5. 

9. 

bu. 

pk. 

qt. 

w.       d. 

h. 

8 

10 

4 

2 
4 

27 

3 

5 
9 

4        6 

13 

7 

8.  If  a  barrel  of  sugar  weigh  2  cwt.  45  lb.  8  oz., 
how  much  will  6  barrels  weigh  ? 

9.  How  much  gold  will  make  a  dozen  rings,  each 
weighing  7  pwt.  15  gr.  ? 

10.  If  a  pupil  study  4  h.  30  min.  each  day,  how 
many  hours  will  he  study  in  12  school  weeks  of  5 
days  each? 

11.  If  a  ship  sail  3^  25'  33^'  in  1  day,  how  far  will 
it  sail  in  15  days? 

12.  If  1  man  can  build  5  rd.  4  yd.  2  ft.  of  fence  in 
a  day,  how  many  yards  can  8  men  build? 

13.  How  much  wheat  in  12  bins,  if  each  bin  con- 
tain  50    bu.  2  pk.  5  qt.? 

14.  John's  age  is  7  yr.  9  mo.  16  d.,  which  is  one- 
fifth  of  the  age  of  his  father:    how  old  is  his  father? 

15.  If  a  load  of  wood  contain  6  cd.  ft.  12  cu.  ft., 
how  much  wood  will  15  loads  make? 

16.  If  a  family  use  2  gal.  3  qt.  1  pt.  of  milk  a  week, 
how  much  will  it  use  in  a  year? 

17.  How  much  hay  is  there  in  6  stacks,  each  con- 
taining 4  T.  16  cwt.  70  lb.? 

/18.  What  is  the  distance  round  a  square  field,  each 
side' of  which  is  24  rd.  3  yd.  2  ft.? 

19.  If  a  printer  use  3  reams  15  quires  12  sheets 
of  paper  each  day,  how  much  paper  will  he  use  in  4 
weeks  of  6  days  each  ? 

20.  If  a  man  walk  2  mi.  7  fur.  32  rd.  an  hour,  how 
far  will  he  walk  in  12  hours? 

21.  A  field  contains  25  rows  of  corn :  if  each  row 
yield  5  bu.  3  pk.,  how  much  corn  will  the  field  yield? 


COMPOUND   NUMBERS.  185 


DEFINITION  AND  EULE. 

Art.  136.  Comjyoiind  Multiplication  is  the  process 
of  taking  a  compound  number  a  given  number  of 
times. 

Art.  137.  EuLE.  —  1.  Write  the  multiplier  under  the 
lowest  denomination  of  the  multiplicand. 

2.  Beginning  at  the  rights  multiply  each  term  of  the 
multiplicand  in  order,  and  reduce  each  product  to  the 
next  higher  denomination,  writing  the  remainder  under 
the  term  multiplied,  and  adding  the  quotient  to  the  next 
product. 

Note. — In  both  simple  and  compound  multiplication  the  suc- 
cessive products  are  each  divided  by  the  numhei'  of  units  of  their  denom- 
ination which  equals  one  of  the  next  higher  denomination. 


LESSON    IV. 

division  of  Cofnpotmd  JV*?imhers. 
1.  Divide  15  w.  G  d.  13  h.  12  min.  by  12. 

PROCESS.  Write  the  divisor   at   the 

12)15  w.  6  d.  13  h.  12  min.  left  of  the  dividend,   as    in 

1  w.  2  d.     7  h.     6  min.  simple    division.     -^^    of    15 

w.  —  1  w..  with  3  w.  remain- 
ing. Write  the  1  w.  under  weeks.  The  3  w.  remaining 
equal  21  d.,  and  21  d.  and  6  d.  equal  27  d.  ^V  ^f  27  d.  — 
2  d.,  with  3  d.  remaining.  Write  the  2  days  under  days. 
The  3  d.  remaining  equal  72  h.,  and  72  h.  and  13  h.  equal 
85  h.  ^  of  85  h.  =  7  h.,  with  1  h.  remaining.  Write  the 
7  h.  under  hours.  The  1  h.  remaining  equals  60  min.,  and 
60  min.  and  12  min.  equal  72  min.  -^  of  72  min.  :=  6  min. 
Write  the  6  min.  under  minutes.  The  quotient  is  1  w.  2  d. 
7  h.  6  min. 


186  INTERMEDIATE   ARITHMETIC. 

(2)  (3)  (4) 

8)14  lb.  12  oz.  15  dr.     10)53  yd.  2  ft.  Sin.     5)52  A.  3  R.  30  P. 

(5)  (6)  (7) 

6)9  cwt.  73  lb.  ;2  oz.    11)65  w.  I  d.  1  h.  58  min.    7)Ulb.  5g.  6.^. 

8.  If  a  inan  sleep  52  h.  30  min.  in  a  week,  how 
long  does  he  sleejD,  on  an  average,  each  day? 

9.  A  man  bought  a  stack  of  hay  containing  0  T. 
19  cwt.  86  lb.,  and  drew  it  home  in  7  equal  loads: 
how  much  hay  did  he  draw  at  each  load? 

10.  If  a   dozen   silver   spoons  weigh   8  oz.  15  pwt., 
what  is  the  weight  of  each  spoon? 

11.  A  farm  of  3-15  A.  3  R  24  P.  was  divided  equally 
between  6  heirs:    how  much  did  each  receive? 

12.  Five   equal   casks   of  vinegar   contain   218   gal. 
2  qt. :    how  much  vinegar  in  each  cask? 

13.  If  a  man  can  dig  a  ditch  36  rd.  4  yd.  2  ft.  long 
in  8  days,  how  much  can  he  dig  in  1  day? 

14.  If  9   men   can    pave    22  sq.  rd.  25  sq.  yd.  in  a 
day,  how  much  can  1  man  pave  in  a  day? 

15.  A  ship  sailed  48°  24'  45''  in  15  days:    how  flir 
did  it  sail  each  day? 

16.  How  many  goblets   can  be  made  of  5  lb.  6  oz. 
12  pwt.  of  silver,  if  each  goblet  weighs  7  oz.  8  pwt.  ? 

PROCESS.  Reduce  both  dividend 

5  lb.  6  oz.  12  pwt.  =  1332  pwt.  and    divisor    to    penny- 

7  oz.     8  pwt.  --    148  pwt.  weights,    and    divide   as 

1332  pwt.  ^  148  pwt.  =  9,   Ans,  "^  '^^^P^^  ^^^^'^^"• 

17.  How  many  bottles,  holding  3  qt.  1  pt.  each,  can 
be  filled  from  a  cask  containing  45^  gallons? 

18.  How  many  baskets  of  peaches,  containing  3  pk. 
4  qt.  each,  will  make  3|  bushels? 


COMPOUND  NUMBERS.  187 

19.  How  many  lengths  of  fence,  each  10  ft.  4  in., 
will  make  28  rd.  3  ft.  of  fence? 

20.  How  many  castings,  weighing  12  lb.  8  oz.  each, 
can  be  made  from  5  cwt.  50  lb.  of  iron  ? 

21.  How  many  times  will  a  wheel,  11  ft.  8  in.  in 
circumference,  revolve  in  going  2  mi.  4  fur.  ? 

22.  How  many  rings,  weighing  5  pwt.  16  gr.  each, 
can  be  made  from  a  bar  of  gold  weighing  1  lb.  8  oz.  ? 

23.  How  many  stej^s,  of  2  ft.  4  in.  each,  will  a  man 
take  in  walking  |  of  a  mile? 

24.  K  a  man  can  walk  2  mi.  6  fur.  in  an  hour, 
how  long  will  it  take  him  to  walk  22  miles? 

DEHNITIOIT  AND  EULES. 

Art.  138.  Compound  Division  is  the  process  of 
dividing  a  compound  number  into  equal  parts. 

Art.  139.  EuLE  I. — 1.  Write  the  divisor  at  the  left 
of  the   dividend,  as   in   simple   division. 

2.  Beginning  at  the  left,  divide  each  term  of  the  divi- 
dend in  order,  and  write  the  quotient  under  the  term 
divided. 

3.  If  the  division  of  any  term  give  a  remainder,  reduce 
it  to  the  next  lower  denomination,  to  the  result  add  the 
number  of  that  denomination  in  the  dividend,  and  then 
divide  as  above. 

Note. —When  the  divisor  is  a  large  number,  the  successive 
terms  of  the  quotient  may  be  written  at  the  right  of  the  dividend, 
as  in  long  division. 

Art.  140.  EuLE  II. — 1.  To  divide  a  compound  num- 
ber by  another  of  the  same  kind,  Beduce  both  com- 
pound numbers  to  the  same  denomination,  and  then 
divide  as  in  simple  division. 


188  INTERMEDIATE   ARITHMETIC. 

LESSON   V. 

Miscellaneotis  Problems. 

1.  If  5  sheets  of  copper  contain  28  lb.  10  oz.  8  dr., 
how  much  copper  is  there  in  each  sheet? 

2.  How  much  silver  will  it  take  to  make  4  dozen 
spoons,  each  spoon  weigiiing  15  pwt.  12  gr.  ? 

3.  If  a  milk  dealer  sell  daily  7  cans  of  milk,  each 
holding  12  gal.  2  qt.,  how  much  milk  does  he  sell 
in  4  weeks? 

4.  From  the  sum  of  15  lb.  8  oz.  15  pwt.  and  9  lb. 
10  oz.  18  pwt.  take  their  difference. 

5.  John  Jones  was  born  Aug.  8,  1856,  and  on 
Jan.  1,  1862,  his  age  was  just  i  of  the  age  of  his 
father:    how  old  was  his  father? 

6.  Two  small  casks,  each  holding  21  gal.  3  qt., 
were  filled  from  a  cask  of  cider  containing  56  gal. 
2  qt. :    how  much  cider  remained  in  the  large  cask? 

7.  A  father  owning  a  ftirm  of  256  A.,  3  R  24  P., 
gave  100  A.  to  his  son,  and  then  divided  what  re- 
mained equally  between  his  two  daughters :  what 
was  each  daughter's  share? 

8.  A  farmer  having  cut  12  T.  15  cwt.  of  hay  from 
a  meadow,  sold  6  loads  of  1  T.  3  cwt.  75  lb.  each, 
and  then  put  the  rest  in  a  stack:  how  much  hay 
was  in  the  stack? 

9.  A  merchant  bought  3  chests  of  tea,  each  weigh- 
ing 2  cwt.  45  lb.,  and  in  one  month  sold  4  cwt. 
80  lb.  12  oz. :    how  much  tea  had  he  left? 

10.  A  publisher  bought  20  bundles  of  paper,  and 
used  daily  3  reams  15  quires  12  sheets:  how  much 
paper  had  he  left  at  the  close  of  10  days? 

11.  A  railroad  company  bought  145  cords  of  wood, 


i 


COMPOUND  NUMBERS.  189 

piled  in  3  ranks;  the  first  rank  contained  36  cd.  5 
cd.  ft.,  and  the  second,  64  cd.  6  cd.  ft.  12  cu.  ft. : 
how  much  wood  was  in  the  third  rank? 
^.^  12.  A  man  bought  3  loads  of  hay,  which,  with  the 
wagon,  weighed  respectively  1  T.  8  cwt.  40  lb. ;  1  T. 
11  cwt.  80  lb.;  and  1  T.  9  cwt.  60  lb.;  and  the  wagon 
alone  weighed  10  cwt.  90  lb. :  how  much  hay  did  he 
buy? 

13.  If  4  horses  eat  15  bu.  3  pk.  4  qt.  of  oats  in  12 
days,  how  much  will  they  eat  in  one  day? 

14.  If  5  horses  eat  21  bu.  1  pk.  6  qt.  of  oats  in 
4  weeks,  how  much  will  3  horses  eat  in  the  same 
time  ? 

15.  How  many  steps,  of  2  ft.  6  in.  each,  will  a  man 
take  in  walking  4  fur.  20  rd.? 

16.  How  many  times  will  a  carriage-wheel,  11  ft. 
4  in.  in  circumference,  turn  round  in  running  10  miles? 

17.  How  many  yards  of  carjieting,  a  yard  wide,  will 
carpet  a  room  18  ft.  by  21  ft.? 

18.  How  many  yards  of  Brussels  carpeting,  |  of  a 
yard  wide,  will  carpet  a  room  20  ft.  by  28  ft.? 

19.  Three  men,  A,  B,  and  C,  bought  a  hogshead 
of  sugar,  weighing  13  cwt.  60  lb. ;  A  received  ^  of  it, 
B  I  of  the  remainder,  and  C  what  was  left:  bow 
much-  sugar  did  each  receive? 

20.  A  company  graded  25  mi.  5  fur.  36  rd.  of  road ; 
^  of  the  job  was  completed  the  first  month,  ^  of  it 
the  second  month,  |  of  it  the  third  month,  and  the 
rest  the  fourth  month:  how  many  miles  of  road 
were  graded  each  month? 

To  Teachers. — See  Manual  of  Arithmetic  for  addi- 
tional  review  problems. 


SECTION   XII. 

TU'M  C  UJVTA  G  U. 


LESSON    I. 

JSTo tat  1071  and  Definitions. 

Art.  141.  The  term  Fer  Cent  means  hy  the  hundred. 

One  per  cent  of  a  number  is  one  hundredth  of  it, 
two  per  cent  is  two  hundredths,  three  per  cent  is  three 
hundredths,  etc. 

1.  How  many  hundredths  of  a  number  is  5  per  cent 
of  it?     7  percent  of  it? 

2.  How  many  hundredths  of  a  number  is  75  per  cent 
of  it?     125  per  cent  of  it? 

3.  How  many  hundredths  of  a  number  is  6^  per  cent 
of  it?     16|  per  cent  of  it? 

4.  How  many  hundredths  of  a  number  is  2|-  per  cent 
of  it?     f  of  one  per  cent  of  it? 

5.  What  per  cent  of  a  number  is  y^^  of  it?  -^-^  of 
it?      125  of  it? 

6.  What  per  cent  of  a  number  is  .06^  of  it?  .24  of 
it?     .331  of  it?     1.12  of  it?     1.45  of  it? 

7.  What  per  cent  of  a  number  is  12^  hundredths  of 
it?     /^%ofit?     .23fofit? 

8.  How  many  hundredths  of  a  number  is  22^  of  it? 
21%  of  it?     63%  of  it?     142%  of  it? 

9.  What  per  cent  of  a  number  is  ^  of  one  hundredth 
of  it?     f  of  one  hundredth  of  it? 

Note.— The  character  %  is  used  instead  of  the  term  "per  cent." 
22  fo  denotes  22  per  cent;  21  fo  denotes  2\  per  cent,  etc. 
(190) 


:  PERCENTAGE.  191 

WRITTEN  EXERCISES. 

10.  Exi)res8  decimally  G  j^er  cent;    12i  per  cent;   | 
of  one  per  cent. 

process:  6fo  =  .06  ;  U^fo  =  .12i;  |^  -^  .OOf. 

11.  Express  decimally  125%;    150%;   200%. 
process:  1257^  =  1.25;  150%  =  1.50;  200%  =  2.00. 

12.  Express  decimally  5%;    7%;    16%;   24%;   40%. 

13.  Express  decimally  i%;    |%;   f%;   ^%;    ^\%. 

14.  Express  decimally .12^%;   16|%;   62^%;   17^^%. 
Suggestion.— 12i%  =  .12i,  or  .125;  62i%  =  .62^,  or  .6225. 

15.  Express  decimally  i%;    f%;    i%;   f%;   ^V%- 

16.  Express  decimally  7j\%i  20^%;  112i%;  1331%. 

DEFINITIONS. 

Art.  142.  Any  Per  Cent  of  a  number  or  quantity  is 
80  many  hundredths  of  it. 

The  term  per  cent  is  a  contraction  of  the  Latin  per  centwiiy 
which  means  by  the  hundred. 

Art.  143.    The    Rate  Per  Cent   is    the   number   of 
hundredths. 

The  character  fo  is^the  per  cent  sign,  and  is  read  per  cent. 

Art.  144.   Percentage  embraces  all  numerical  opera- 
tions in  which  one  hundred  is  the  basis. 

LESSON    II. 
Case  L— To  Find  a  Given  Per  Cent  of  any  Number. 

1.  How  much  is  5  per  cent  of  200? 
Solution.  —  b%  of  200  is  ^^  of  200 :  -^  of  200  is  2,  and 
r§TT  of  200  is  5  times  2,  which  is  10:   5%  of  200  is  10. 
Or,  5^  of  200  is  ^  of  200,  which  is  10. 


192  INTERMEDIATE  ARITHMETIC. 

2.  Wluit  is     3  per  cent  of  400?     8%  of  500? 

3.  What  is     6  per  cent  of  150?     4%  of  250? 

4.  What  is  10  per  cent  of  $900?     15%  of  $600? 

5.  What  is  8%  of  2000?   12%  of  4000?   3%  of  2500? 

To  Teachers.— Show  that  Sfo  of  2500  may  also  be  found 
by  multiplying  2500  by  .03,  and  then  require  the  pupils  to 
solve  the  above  problems  in  this  manner. 

WRITTEN  EXERCISES. 

6.  What  is  16%  of  324?   "51%  of  $724.50? 

324 

.16 

process: r 

1944 

324 


51.84,  Am. 

7.  What  is  8%  of  $3250? 

8.  6  %  of  245? 

9.  9  %  of  1200? 
10.-  15  %  of  644? 

11.  5   %  of  1540? 

12.  10  %   of  1050  ft.? 

13.  33  %  of  560  lb.? 

14.  31%  of  321  oz.  ? 

15.  12|%  of  960  men? 

24.  A  man  has  an  income  of  $2540,  and  his  expenses 
are  62i  per  cent  of  his  income:  how  much  are  his  ex- 
penses ? 

25.  A  man  having  285  acres  of  land  gave  334^%  of  it 
to  his  daughter:  how  many  acres  had  he  left? 

26.  A  drover  bought  245  sheep  of  A,  and  60%  as 
many  sheep  of   B  :    how  many  sheep  did  he  buy? 

27.  A  ship   is  valued   at   $15800,  and   the  cargo  is 


$724.50 

.05i 

SSS  I 

24150 

362250 

$38.6400,  Ans. 

16. 

i%  of  $450? 

17. 

f  %  of  $525? 

18. 

1  %  of  365  days? 

19. 

^%  of  $9650? 

20. 

6%   of  $.621? 

21. 

^  of  6.45? 

22. 

3%  of  40.5  ft.? 

23. 

6-1  %  of  96.6  miles? 

PERCENTAGE.'  193 

worth   15^   less  than  the  ship:  what  is  the  value  of 
the  cargo? 

28.  An  army  of  8450  men  lost  22%  of  its  men  in 
battle:  how  many  men  did  it  lose? 

29.  A  school  enrolled  320  pupils  in  a  term,  and  the 
average  number  in  daily  attendance  was  82^%  of  the 
number  enrolled:  what  was  the  average  number  of 
pupils  in  daily  attendance  ? 

30.  A  farmer  raised  2450  bushels  of  grain,  and  42% 
of  the  grain  was  wheat,  24%  oats,  and  the  rest  corn: 
how  many  bushels  of  each  kind  of  grain  did  he  raise? 

Art.  145.  EuLE. — To  find  a  given  per  cent  of  any 
number,  Multiply  the  number  by  the  given  rate  per  centy 
expressed  decimally. 

Note. — When  the  rate  is  an  aliquot  part  of  100,  the  per  cent 
may  be  found  by  taking  the  same  aliquot  part  of  tlie  number. 
Thus,  33i%  of  $48  is  i  of  $48;  12hfo  of  320  is  h  of  320;  25%  of  84 
is  i  of  84. 

LESSON  III. 

Case  IL— To  Find  what  Per  Cent  one  Number  is 
OF  Another. 

1.  What  per  cent  of  12  is  3? 

Solution.— 1  is  yV  of  12,  and  3  is  x\,  or  ^  of  12;  J  is  .25, 

or  25%.     3  is  25%  of  12. 

2.  What  per  cent  of  75  is  15? 

3.  120  is  (^0?  5.  90  is  15?  7.  320  is  32? 

4.  125  is  25?  6.  72  is  36?  8.  128  is  16? 

To  Teachers. — Show  that  the  rate  per  cent  in  each  of  the 
above  problems  may  also  be  found  by  dividing  the  number 
which  is  the  percentage   by  the  other  number.     Thus:  (1)  j\  = 

3  ^  12  :=  .25  =.  2hcJo\  (8)  ^2-8  =-  16  -^  128  ^  .125  ==  12^%. 
T.  A.— 13. 


19^  INTERMEDIATE    ARITHMETIC. 

WRITTEJJ   EXERCISES. 

9.  What  per  cent  of  $520  is  $23.40? 

$520)$23.40(.04r)  =  4^;^,  Ans, 

2080 
PKOCESS:  -j^^ 

2600 


10.  What  per  cent  of  75  is  13.5? 

11.  640  is  48?  18.  $324  is  $356.40? 

12.  $650  is  $32.50?  19.  $324  is  $32.40? 

13.  38  lb.  is  5.32  lb.?  20.  $3.20  is  $3.36? 

14.  900  yd.  is  112.5  yd.?  21.  $2.40  is  $2.04? 

15.  $128  is  $5.76?  22.  12|-  cts.  is  5  cts.? 

16.  $7.20  is  $1.08?  23.  12|  cts.  is  10  cts.? 

17.  $392.50  is  $3.92|?  24.  4^  is  |? 

25.  A  drover  bought  45  horses,  and  sold  18  of  them : 
what  per  cent  of  the  drove  did  he  sell? 

26.  A  merchant  bought  432  yards  of  silk,  and  sold 
288  yards :  what  per  cent  of  the  silk  did  he  sell? 

27.  A  school  enrolled  225  pupils  in  a  term,  and  the 
average  number  in  daily  attendance  was  198 :  what 
was  the  per  cent  of  attendance? 

28.  An  army  of  15450  men  lost  1236  men  in  battle : 
what  per  cent  of  the  army  was  lost? 

29.  If  3740  pounds  of  ore  contain  2618  pounds  of 
iron,  what  per  cent  of  the  ore  is   iron? 

30.  A  man  having  an  income  of  $2750  pays  $440 
rent,  and  $1650  for  other  expenses:  what  per  cent  of 
his  income  has  he  left? 

Art.  146.  EuLE.— To  find  what  per  cent  one  num- 
ber is  of  another,  Divide  the  number  which  is  the  per- 
centage by  the  other  number,  and  the  quotient,  expressed 
as  hundredths,  will  be  the  rate  per  cent. 


PERCENTAGE.  195 

LESSON    IV. 

Case  III. — To  Find  a  Number  when  a  Per  Cent  of 
IT  IS  Given. 

1.  60  is  20%   of  what  number? 

Solution. — If  60  is  20^  of  a  number,  1%  of  it  is  ^V  of  60, 
which  is  3,  and  100^,  or  the  number,  is  100  times  3,  which  is 
300 :   60  is  20  fo  of  300. 

2.  50  is  25%  of  what  number? 

3.  75  is  15%  of  what  number? 

4.  $36  is  3%  of  how  many  dollars? 

5.  $70  is  10%  of  how  many  dollars? 

6.  240  acres  are  12%  of  how  many  acres? 

7.  230  miles  are  115%  of  how  many  miles? 

8.  500   rods   are  125%  of  how  many  rods? 

To  Teachers. — Show  that  the  result  obtained  above  by- 
analysis  may  also  be  obtained  by  dividing  the  given  number 
by  the  rate  per  cent  expressed  decimally.  Thus:  (1)  60  -^  20 
X  100  =  60  -f-  .20  ==  300 ;  (8)  500  rd.  ^  125  X  100  =  500  rd. 
-4-  1.25  =  400  rd. 

WRITTEN   EXERCISES. 

9.  A  man  owes  $3175,  which  is  12|^%  of  his  estate : 
what  is  the  value  of  his  estate? 

process:  $3175 -V- .125  =  $25400,  Am, 

10.  75  is  37|%  of  what  number? 

11.  $23.10  is  5%  of  what  sum  of  money? 

12.  $61.60  is  110%   of  what  sum  of  money? 

13.  $8.16 Js  121%  of  what  sum  of  money? 

14.  $180  is  831%  of  what  sum  of  money? 

15.  A  lady  paid  $31.50  for  a  chain,  which  was  45% 
of  what  she  paid  for  a  watch :  what  was  the  cost  of 
the  watch? 


196  INTERMEDIATE  ARITHMETIC. 

16.  A  farmer  sold  420  pounds  of  wool,  which  was    | 
15%   of  his  clip:  how  much  wool  did   he  shear? 

17.  A  man  invested  $4050,  which  was  7^%  of  his 
property:   what  was  tiie  value  of  his  property?  1 

18.  A  man  paid  $375  a  year  for  a  house,  $1275  for 
other  expenses,  and  laid  up  25%  of  his  income:  what 
was  his  income? 

Art.  147.  EuLE. — To  find  a  number  when  a  per  cent 
of  it  is  given,  Divide  the  number  which  is  the  percentage 
by  the  rate  per  cent,  expressed  decimally, 

LESSON    V. 
Profit  and  Loss, 

1.  A  man  paid  $80  for  a  horse,  and  sold  it  for  10% 
more  than  it  cost  him :  for  how  much  did  he  sell  it  ? 

Solution.— 10/^  of  $80  is  $8,  and  $80  plus  $8  is  $88:  he 
sold  it  for  $88. 

2.  A  man  bought  a  horse  for  $80,  and  sold  it  for  $88: 
Avhat  per  cent  did  he  gain  ? 

Solution.— If  he  gave  $80,  and  sold  for  $88,  he  gained  $88 
less  $80,  which  is  $8 ;  $8  is  f^,  or  ^^  of  the  cost,  and  ^V  equals 
.10,  or  10^  :  he  gained  10  per  cent. 

3.  A  man  sold  a  horse  for  $88  and  gained  10%:  what 
was  the  cost  of  the  horse? 

Solution.— Since  he  gained  10%,  $88  is  110%  of  the 
cost;  if  $88  is  110%,  1%  is  $.80,  and  100%  is  $80:  the 
horse  cost  $80. 

4.  A  dealer  paid  $5  for  a  hat,  and  sold  it  at  20% 
profit:  what  was  the  selling  price? 

5.  A  dealer  paid  $5  for  a  hat,  and  sold  it  for  $G: 
what  per  cent  did  he  gain? 


PERCENTAGE.  197 

G.  A  dealer   sold  a  hat   for   $G   and  gained   20^  : 
what  was  the  cost  of  the  hat? 

7.  For  how  much  must  silk,  that  cost  $1.20  a  yard, 
be  sold  to  gain  25%? 

8.  Silk  that  cost  $1.20  a  yard  was  sold  for  $1.50: 
what  was  the  gain  per  cent? 

9.  Silk  was  sold  for  $1.50  a  yard,  at  a  gain  of  25%  : 
what  was  the  cost? 

10.  For  how  much  must  butter,  that  cost  20  cents 
a  pound,  be  sold  to  gain  10%? 

11.  A   carriage   that   cost   $150  was   sold   for  $120: 
w^hat  was  the  per  cent  of  loss? 

12.  A   merchant   bought  velvet   at   $4   a   yard,  and 
sold   it  at  $5   a  yard:  what  per  cent  did  he  gain? 

13.  A   merchant  bought  velvet   at  $5   a   yard,   and 
sold  it  at  $4:  what  was  the  per  cent  of  loss? 

14.  A  merchant  sold  velvet  at  $4  a  yard,  and  lost 
20%:  what  was  the  cost? 

"WKITTEN    EXEKCISES. 

15.  A  man  paid  $8750  for  a  farm,  and   sold   it  at 
40%  profit;  what  was  the  selling  price? 

16.  A  man  sold  a  farm  for  $12250,  and  gained  40%: 
what  was  the  cost? 

PROCESS:  $12250  --  1.40  ==  $8750,  the  cost. 

17.  A  cargo  of  wheat,  which  cost  $24650,  was  sold 
at  a  loss  of  12%:  for  how  much  was  it  sold? 

18.  A  drover  paid  $135  a  head  for  horses,  and  sold 
them  at  30%  profit:  what  was  the  selling  price? 

19.  A   house   that   cost   $3840  was   sold   for   $4128: 
what  was   the   gain    per   cent? 

20.  For  how  much   must  teas,  that  cost  $.90,  $1.05, 
and   $1.10  a  pound,  be  sold  to  gain   20%? 


198  INTERMEDIATE  APaTHMETIC. 

LESSON    VI. 
Comtmssiony  Disurance^    Taxes,  etc. 

TATRITTEN  EXERCISES. 

1.  An  agent  sold  a  farm  for  $3500,  and  received  a 
commission  of  5^:    how  much  did  he  receive? 

Note. — The  teacher  should  explain  the  business  terms  used  in 
these  problems,  and  add  such  information  as  will  be  of  value  to 
the  pupil. 

2.  An  attorney  collected  a  debt  of  $324.50,  and 
charged  10%  for  his  services:  what  was  his  com- 
mission? 

3.  An  architect  furnished  plans  and  superintended 
the  erection  of  a  building  for  2^%  of  its  cost,  which 
was  $25400:  how  much  did  he  receive? 

4.  A  commission  merchant  sold  320  barrels  of  flour 
at  $6.50  a  barrel,  and  charged  \\%  commission:  how 
much  did  he  receive? 

5.  A  broker  bought  $12300  worth  of  cotton,  and 
charged   i%:    what  was    his    commission? 

6.  A  store  worth  $25600  is  insured  for  f  of  its 
value,   at  \%\  what   is   the   premium? 

ff  of  $25600  =  $19200,  amount  insured. 
process:  { 

($19200  X  .005 -.$96,  premium. 

7.  A  house  worth  $7500  is  insured  for  |  of  its  value, 
at  f%:  what  is  the  premium? 

8.  A  man  has  his  life  insured  for  $4000,  at  $23.50 
per  $1000:  what  annual  premium  does  he  pay? 

9.  A  man's  property  is  listed  at  $10450,  and  the  tax 
levy  is  15  mills  on  the  dollar:   what  is  his  tax? 

process:     $10450  X  .015  =  $156.75,  tax. 


PERCENTAGE.  199 

10.  A  man  j^ays  a  tax  of  12|^  mills  on  property 
listed  at  $4960 :    what  is  his  tax  ? 

11.  The  taxable  property  in  a  certain  village  is 
listed  at  $316000:  if  a  tax  of  $4740  is  assessed  to  build 
a  school-house,  what  will  be  the  rate  in  mills? 

12.  A  merchant  imported  a  lot  of  silk  invoiced  at 
$32600:  what  was  the  duty,  at  37^%? 


LESSON    VII. 

Simple  Tnte7^est. 

Art.  148.  Interest  is  money  paid  for  the  use  of 
money. 

The  Vrinripal  is  the  sum  of  money  for  the  use 
of  which  interest  is  paid. 

The  Amount  is  the  sum  of  the  principal  and  the 
interest. 

Art.  149.  Simple  Interest  is  interest  on  the  princi- 
pal only. 

Note.— See  Complete  Arithmetic  for  methods  of  computing 
Annual  Interest  and  Compound  Interest. 

The  Six  Per  Cent  Method. 

Art.  150.  When  money  is  loaned  at  6  per  cent  per 
annum,  the  interest  of  $1  for  1  year  is  .06  of  $1,  or 
6  cents. 

1.  What  is  the  interest  of  $1   for  2  years,  at  6%? 

Solution. — Since  the  interest  of  $1  for  1  year,  at  6%,  is 
6  cents,  the  interest  of  $1  for  2  years  is  2  times  6  cents,  which 
is  12  cents. 

2.  What  is  the  interest   of  $1    for  4  years,  at  6%? 


200  INTERMEDIATE   ARITHMETIC. 

3.  What  is  the  interest  of  $1   for  5  years,  at  6%  ? 
7  years?     9  years?     10  ^^ears? 

4.  What  is  the  interest  of  $1  for  1  month,  at  6%  ? 

Solution. — Since  the  interest  of  $1  for  12  months  is  6 
cents,  the  interest  for  1  month  is  ^^  of  6  cents,  which  is  J 
cent,  or  5  mills. 

5.  What  is  the  interest  of  $1  for  3  months,  at  G%  ? 

Solution. — Since  the  interest  of  $1  for  one  month,  at  6%, 
is  5  mills,  the  interest  for  3  months  is  3  times  5  mills,  which  is 
15  mills. 

6.  What  is  the  interest  of  $1  for  4  months,  at  6%  ? 

7.  What  is  the  interest  of  $1  for  5  months,  at  6%  ? 
7  months?     9  months?     11  months? 

8.  What  is  the  interest  of  $1  for  2  years  4  months, 
at  G%? 

Solution. — The  interest  for  2  years  is  12  cents,  and  for  4 
months  2  cents;  12  cents  and  2  cents  are  14  cents:  the 
interest  of  $1  for  2  yr.  4  mo.  is  14  cents. 

9.  What    is   the   interest  of  $1    for   3   yr.   6   mo.? 
For  1  yr.  10  mo.?     2  yr,  8  mo.? 

10.  What  is  the  interest  of  $1  for  2  yr.  5  mo.? 
3  yr.  7  mo.?     4  yr.  3  mo.?     1  yr.  1  mo.? 

11.  What  is  the  interest  of  $1  for  6  days,  at  6%? 

Solution.— Since  the  interest  of  $1  for  1  month,  or  30 
days,  is  5  mills,  the  interest  for  1  day  is  ^^o  ^^  ^  mills,  which 
is  J  mill,  and  the  interest  for  G  days  is  6  times  J  mill,  which 
is  1  mill. 

12.  What  is  the  interest  of  $1  for  12  days,  at  G%? 

Solution. — Since  the  interest  of  $1  for  6  days  is  1  mill, 
the  interest  for  12  days,  or  2  times  6  days,  is  twice  1  mill, 
which  is  2  mills. 


PERCENTAGE.  201 

13.  What  is  the  interest  of  $1  for  24  days,  at  0%? 
18  days?     9  dajs?     21  days? 

14.  What  is  the  interest  of  $1  for  21  daj^s  at  6%? 
8  days?     14  days?     20  days? 

15.  What   is   the  interest   of  $1   for  2  mo.   18   da.? 

4  mo.  24  da.?     3  mo.  6  da.? 

16.  What   is  the   interest   of  $1    for   2    mo.    9    da.? 

5  mo.  12  da.?     5  mo.  8  da.?     1  mo.  1  da.? 

WRITTEN   EXERCISES. 

17.  What  is  the  interest  of  $1  for  2  yr.  3  mo.  14  da., 

at  6%? 

Int.  of  $1  for  2  yr.  =  $.12 
''     "     ''     ''     3  ino.-^    .015 
"     "     ''     "  14  da.  ^    .0021 

$.1371,  A71S. 

What  is  the  interest  of  $1,  at  6%,  for 

18.  1  yr.  9  mo.  6  da.?  22.  2  yr.    2  mo.  2  da.? 

19.  2  yr.  1  mo.  25  da.?  23.  1  yr.    1  mo.  1  da.? 

20.  3  yr.  3  mo.  20  da.?  24.  7  mo.  7  da.? 

21.  1  yr.  1  mo.  24  da.?  25.  5  jr.    5  da.? 

26.  What   is   the   interest   of  $425    for  2  yr.   3  mo. 
12  da.,  at  6%? 


Since  the  intere.st  of  $1 
for  2  yr.  3  mo.  12  da.,  at 
6fo,  is  $.137,  or  .137  of  $1, 
the  interest  of  $425  is  .137 
of  $425,  which  is  $58,225. 


27.  What  is  the  interest  of  $145.60  for  1  yr.  5  mo. 
24  da.,  at  6%  ?     For  2  yr.  2  mo.  9  da.? 


Int.  of  $1  =  $.137. 

$425 

.137 

DESS:             2  97  5 

1275 

425 

$5  8,2  2  5,  Int. 

202  INTERMEDIATE   ARITHMETIC. 

28.  What  is  the   interest  of  $64.20  for  1  yr.  3  mo., 
at  6%?     For  5  mo.  15  da.? 

29.  What  is  the   amount  of  $85.50  for  1   yr.  1  mo. 
1  da.,  at  6%?     For  4  yr.  10  da.? 

Note. — The  amount  is  the  sum  of  principal  and  interest. 

30.  What  is  the  amount  of  $184.80  for  9  mo.  27  da., 
at  6%?     2  yr.  25  da.? 

31.  Wliat  is  the  interest  of  $31.20  from  Oct.  23,  1855, 


$31.20 
.208^ 
520 
24960 
6240 


I 


to 

Apr.  12^ 

1859, 

at  8%? 

1859 

4 

12 

1855 

10 

23 

Syr. 

5  mo.  1  9  da. 

In 

t.  of$lat6%  =  $.208;. 

8)$6.49480,  Int.  at  6%. 
2.16493,     ''     ''   1%, 
$8.65973,     "     "   8^. 

32.  What   is  the  interest  of  $540.50  from  May  10, 

1873,  to  March  4,  1874,  at  7%  ?     At  9%  ? 

Suggestion.— The  interest  at  7%  is  \  more  than  the  interest  at 
6%,  and  the  interest  at  9%  is  J  more  than  the  interest  at  6^^  . 

33.  Wliat  is  the   amount  of  $121.60  from  Feb.   12, 

1874,  to  May  22,  1876,  at  8%  ?     At  12%? 

'  34.  A  man  borrowed  $460,  July  28,  1866,  and  paid 
it  May  16,  1869,  with  interest  at  5%:  what  was  the 
amount? 

35.  A  note  of  $243.75,  dated  June  8,  1873,  was  paid 
Nov.  14,  1875,  with  interest  at  10%  :  what  was  the 
amount  ? 

36.  A  note  of  $600,  dated  Sept.  9,  1872,  was  paid 
Jan.  21,  1874,  with  interest  at  7.3%  :  what  was  the 
amount? 


PERCENTAGE.  203 

Art.  151.  EuLES. — 1.  To  comijute  interest  at  6  per 
cent,  Find  the  interest  of  ?1  for  the  given  time^  and 
then  multiply  the  given  principal  by  the  abstract  deci- 
7nal  which  corresponds  to  the  interest  of  %\. 

Note. — The  interest  of  $1  may  be  found  by  taking  six  times  as 
many  cents  as  there  are  years,  one- half  as  many  ceiits  as  there  are 
months,  and  one-sixth  as  many  mills  as  there  are  days. 

2.  To  compute  interest  at  any  rate  per  cent,  Find 
the  interest  at  6  per  cent^  and  then  increase  or  diminish 
this  interest  by  such  a  part  of  itself  as  will  give  the  in- 
terest at  the  given  rate. 

Notes. — 1.  The  following  table  may  be  found  convenient: 


7  % 

=  6fo 

+   i    of  (yfo. 

12  ^0   =  6fo 

X  2. 

n% 

--    (jfc 

+  i   of  Qfc, 

15  fc   ^  6fo 

X  2J. 

8  % 

-=  Qfo 

+  1   of  6fc. 

10  %    =  6% 

-^-  6  X  10. 

9  % 

=  G% 

+  i   of  6%. 

11    fo    =   Qfo 

-^-  6  X   11. 

5  /. 

=  6fc 

—  i   of  Gfo. 

7.3  fo   =  Qfo 

-f-  6  X  7.3 

4  fo 

=   Gfo 

—  1   of  Qfo, 

bifo    =  Qfo 

-  6  X  5i. 

^fo 

=  6fo 

—  i  of  efo. 

X   fo    =    6fo 

--  6  X  ^. 

2.  The  interest  at  10%  is  found  by  dividing  the  interest  at  6% 
by  6  and  removing  the  decimal  point  one  place  to  the  right. 

3.  The  interest  of  $1  at  7.3%  is  1  mill  for  5  days,  and  hence 
interest  for  any  number  of  days,  at  7.3%,  (for  365  days,)  may  be 
computed  by  multiplying  the  principal  by  |  as  many  thousandths  as 
there  are  days  in  the  time.  The  answer  to  problem  36,  p.  202,  is 
$659.88,  if  the  interest  be  computed  for  the  actual  number  of  days. 


LESSON    VIII. 

!2)fscou7it. 

Art.  152.   jyiscount  is  a  deduction  from  a  debt  for 
its  payment  before  it  is  due. 

A  note  due  at  a  future  date  without  interest  is  usually 
discounted   in    business   by   deducting    the    interest    for   the 


204  INTERMEDIATE   ARITHMETIC. 

time,  with  or  without  grace,  as  per  agreement.  The  rate 
of  interest  allowed  is  usually  greater  than  the  current  rate. 
Bills  are  often  discounted  by  deducting  a  certain  per  cent 
of  their  face  without  regard  to  time.  These  deductions  are 
^called  Business  Discount. 

Deductions  from  the  nominal  price  of  articles  sold  are 
often  computed  as  a  certain  per  cent  off.  This  deduction  is  called 
Trade  Discount. 

Note.  —  The  deduction  known  as  *' True  Discount^*  is  seldom 
made  in  business,  and  hence  is  not  presented  in  this  book.  It 
is  fully  treated  in  the  Complete  Arithmetic  {p.  192). 

Art.  153.  Sank  Discount  is  the  interest  on  a  note 
paid  in  advance. 

When  a  bank  loans  money,  the  borrower  gives  his  note, 
payable  at  a  specified  time  without  iriterest.  The  interest  on 
this  note  for  the  time,  plus  three  days,  is  subtracted  from  its 
face,  and  the  remainder,  called  the  proceeds^  is  paid  to  the 
borrower.  The  three  days  added  to  the  time  are  called  Days 
of  Grace. 

WRITTEN    EXERCISES. 

1.  A  bank  discounted  a  note  of  $250,  payable  in 
60  days,  at  8%  :    what  were  the  proceeds? 

PROCESS. 

$250 
.0105 


60  days  +  3  days  ==  63  days.  12  5  0 

Int.  of  $1  at  6^  =  $.0105.  ^^^ 

3)$2.6250,   Discount  at  6^. 

8750 

$3.50,         Discount  at  8%. 
$250  — $3.50  ==$246.50,  Proceeds. 

2.  What  are  the  proceeds  of  a  note  of  $240.60,  pay- 
able in  90  days,  discounted  by  a  bank  at  10%  ? 


PERCENTAGE.  205 

What  are  the  bank  proceeds  of  a  note  of 

3.  $22.50,  payable  in  60  days,  discounted  at  9%  ? 

4.  $720,  payable  in  30  days,  discounted  at  8%  ? 

5.  $62.40,  payable  in  45  days,  discounted  at  10%? 

6.  $125,  payable  in  90  days,  discounted  at  7|%  ? 

7.  What  are  the  proceeds  of  a  note  of  $90.60,  dated 
March  10,  1876,  and  payable  June  5,  1876,  discounted 
by  a  bank  at  9%  ? 

PROCESS. 

In  March,  21  days.  $90.6  0 
"  April,    30     -  .015 

"May,      31     "  "J^^^ 

"  June,       5     "  9060 

^'''!\-^  2)11.35900, 

6  )  90  days.  ^^^r^ 

Int.  of  $1  at  6fe  =  15  mills.  "^oAoor    t^»       .  ^^ 

^  $2.0  3  8  5,  Dis.  at  d%, 

$90.60  — $2.0385  =  $87,561  5,  Proceeds. 

Note. — When  the  time  of  a  note  is  short,  it  is  the  general  cus- 
tom of  bankers  to  compute  interest  for  the  actual  number  of  days 
in  the  time,  inchiding  grace,  each  day  being  considered  as  ^J^  of  a 
year.  This  mode  of  finding  the  time  of  interest,  with  or  without 
grace,  is  called  the  Method  by  Days. 

8.  What  are  the  proceeds  of  a  note  of  $142,  dated 
Aug.  15,  1875,  and  payable  Nov.  4,  1875,  discounted 
at  10%  ? 

9.  A  note  of  $360,  dated  Sept.  15,  1875,  and  pay- 
able Nov.  15,  1875,  was  discounted  by  a  bank  at  8%  : 
what  were  the  proceeds? 

10.  A  note  of  $225,  dated  May  11,  1875,  and  payable 
July  31,  1875,  with  interest  at  6%,  was  discounted 
June  4,  1875,  at  10%:   what  were  the  proceeds? 

Suggestion. — Compute  the  interest  for  81  days  -[-  3  days,  or  84 
days,  and  discount  the  amount  thus  found  for  60  days. 


206  INTERMEDIATE  ARITHMETIC. 

11.  A  merchant  discounted  a  bill  of  $750.  due  in 
3  months,  by  deducting  the  interest  for  the  time, 
without  grace,  at  8%  :   what  were  the  cash  proceeds? 


PROCESS. 

Int.  of  $1  for  3  mo.  at  6^  = 

:  $.015 

$750 
.015 

8)$11.250,  Dis.  at  6/.. 
3  75 

$15.00,     Dis.  at  8^. 

$750  —  $15  =  $735,  Cash  proceedf?. 

12.  A  man  sold  a  note  of  $150,  due  in  6  months, 
at  a  discount  of  10^  for  the  time,  without  grace: 
how  much  did  he  receive  for  the  note? 

13.  A  merchant  deducted  for  cash  5%  from  a  bill 
of  goods  amounting  to  $540:  how  much  did  he  re- 
ceive for  the  goods? 

process:     $5.40  X  .05  -=  $27.      $540  — $27  =^  51  3,  Ans. 

14.  A  man  asked  $12500  for  a  farm,  but  sold  it 
for  6%  off  for  cash  down:  what  did  he  receive  for 
the  farm? 

15.  A  man  bought  a  bill  of  goods  amounting  to 
$1500  on  60  days'  credit,  but  was  offered  a  discount 
of  5%  for  cash:   how  much  did  he  pay  for  the  goods? 

Art.  151.  EuLES.  —  1.  To  compute  bank  discount. 
Find  the  interest  of  the  sum  discounted  for  the  number 
of  days  in  the  time  plus  three  days. 

2.  To  find  the  proceeds,  Subtract  the  discount  from 
the  sum  discounted. 


PERCENTAGE.  207 


LESSON    IX. 

JVbteSy  Drafts,  and  !Bonds, 

Art.  155.  A  Proniissorif  Note  is  a  written  agree- 
ment by  one  person  to  pay  another  a  specified  sum 
of  money  at  a  specified  time. 

The  sum  of  money  specified  is  called  the  Face  of  the 
note. 

The  person  who  signs  a  note  is  its  Maker ;  the  person 
to  whom  it  is  payable  is  the  Payee;  and  the  owner  is 
the  Holder. 

An  Indorser  is  a  person  who  signs  his  name  on 
the  back  of  a  note  as  security  for  its  payment. 

Art.  156.  A  Draft  is  an  order  by  one  person  upon 
another  to  pay  a  specified  sum  to  a  third  person 
named.     It  is  also  called  a  Bill  of  Exchange, 

The  process  of  making  payments  at  distant  places  by  the 
remittance  of  drafts  is  called  Exchange. 

Art.  157.  Bonds  are  interest  bearing  notes  issued 
by  nations,  states,  cities,  railroad  companies,  and  other 
corporations,  as  a  means  of  borrowing  money.  They 
are  issued  under  seal. 

The  market  value  of  bonds  is  quoted  at  a  certain  per  cent 
of  their  face,  or  par  value.  Bonds  quoted  at  109  are  worth, 
in  currency,  109^  of  their  face. 

To  Teachers.  —  For  fuller  information  respecting  notes, 
drafts,  and  bonds,  see  Complete  Arithmetic,  pp.  198-202, 
and  311. 


208  INTERMEDIATE   ARITHMETIC. 

A^TBITTEN   EXERCISES. 

1.  What  will  be  the  cost  of  a  draft  on  New  York 
for  $640,  when  exchange  is  f %   premium? 

$640  X  .00J=-$2.40,  Cost  of  exchange. 


process:   , 

$640  +  $2.40  =  $642.40,  Cost  of  draft. 

2.  A  merchant  in  St.  Louis  wishes  to  remit  $2450 
to  a  creditor  by  draft  on  New  York:  what  will  be 
the  cost  of  the  draft,  at  f%  premium? 

3.  A  merchant  in  New  Orleans  bought  a  sight  draft 
on  New  York  for  $5600,  at  |%  discount: .  what  was 
the  cost  of  the  drafl? 

4.  When  gold  was  quoted  at  112 J,  what  was  the 
value  in  currency  of  $5440  in  gold  ? 

PROCESS :   $5  440X1-125  =  $6120,  Value  in  currency. 

Note. — When  gold  is  quoted  at  112K,  it  is  worth  112K^,  or 
12>^  more  than  currency;  that  is,  $1  in  gold  is  worth  $1.12^  in 
currency,  and  $100  in  gold  is  worth  $112.50  in  currency.  Gold 
has  not  been  at  a  premium  and  thus  quoted  since  1878. 

5.  When  gold  was  quoted  at  112^,  what  was  the 
value  in  gold  of  $6120  in  currency? 

process:   $6  1  20^1.1  25:=:=$5440,  Value  in  gold. 

6.  When  gold  was  quoted  at  112,  what  was  the  value 
in  currency  of  $1752.50  in  gold? 

7.  When  gold  was  quoted  at  112,  how  large  a  gold 
draft  could  be  bought  for  $196280  in  currency? 

8.  When  gold  was  quoted  at  110,  what  was  the  gold 
value  of  a  one  dollar  bank  note? 

9.  When  gold  was  quoted  at  112^,  what  was  the 
value  in  currency  of  a  five  dollar  gold  piece? 

10.  What  was  the  cost  in  currency  of  a   gold   draft 
for  $500,  when  gold  was  quoted  at  $110? 


SECTIOK   XIII. 


j. — I — i 

I 1 4 


LESSON    I. 
Su7''faces* 

1.  How  many  square  feet  in  a  piece  of  zinc,  5  feet 
long  and  3  feet  wide? 

PROCESS :    5  X  3  =  1 5,  sq.  ft. 

In  a  piece  5  feet  long  and  1  foot  wide 
there  are  5  sq.  ft.,  and  in  a  piece  3  feet 
wide  there  are  3  times  5  sq.  ft.,  which 
is  15  sq.  ft. 

2.  How  many  acres  in  a  field  45  rods  long  and  32 
rods  wide? 

3.  How  many  square  inches  in  a  triangle  whose  base 
is  15  inches  and  altitude  10  inches. 


PROCESS. 

15  X  10  —  150,  sq.  in.  in  rectangle. 
150  -r-  2  =  75,  sq.  in.  in  triangle. 

The  area  of  a  triangle  is  one  half 
of  the  area  of  a  rectangle  with  the 
same  base  and  altitude. 


Base,  15  inches. 


4.  How  many  square  yards  in  a  triangular  surface 
with  a  base  of  18  feet  and  an  altitude  of  12  feet? 

5.  How  many  square  feet  in  a  board  14  feet  long 
and  8  inches  wide  at  one  end  and  4  inches  wide  at 
the  other? 

I.  A.— 14.  (209) 


210  INTERMEDIATE  ARITHMETIC. 

Note. — The  average  width  of  the  board  is  i  of  the  sum  of  8 
inches  and  4  inches,  which  is  6  inches,  or  i  foot.  In  j)ractice, 
this  would  be  found  by  measuring  the  width  of  the  board  at  its 
middle. 

6.  How  many  square  feet  of  lumber  in  15  boards, 
each  12  ft.  long  and  7  in.  wide,  and  in  21  boards, 
each  14  ft.  long  and  8  in.   wide? 

7.  How  many  square  feet  of  timber  in   12  joists, 

2  in.    by   8   in.,   and    15   ft.   long,    and    20    scantling, 

3  in.   by  4   in.,   and   12   ft.   long? 

8.  The  diameter  of  a  circle  is  5  inches :  how  many 
inches  in  its  circumference? 

PROCESS.  y^^^^  \  ^^X,^^ 

5  in.  X  3.1416  =  15.708  in.,  circum-         /             I             \ 
ference.  [ '^ - 

It  may  be  shown  by  geometry  that         \  / 

the  circumference  of  a  circle  is  3.1416              \^^  / 

(nearly  3f)  times  the  diameter.  

9.  What  is  the  circumference  of  a  wheel  whose 
diameter  is  15  feet? 

10.  A  circular  room  is  40  feet  in  diameter:  how 
many  square  feet  in  the  floor? 

PROCESS. 

4  0  ft.  X  3.1416^  125.664  ft,  circumference. 
125.664X40-4-4  =  1256.64,  sq.  ft.  in  the  floor. 

It  may  be  proven  by  geometry  that  the  area  of  a  circle  is 
equal  to  the  circumference  multiplied  by  one-fourth  of  the 
diameter,  or  One-half  of  the  radius. 

11.  A  horse  is  tied  to  a  stake  by  a  rope  40  feet 
long:    on  how  much  surface  can  the  horse  graze? 

Suggestion.  —  The  surface  is  a  circle  80  feet  in  diameter. 


MENSURATION.  211 

12,  How  many  square  inches  in  the  surface  of  a 
ball  3  inches  in  diameter? 

PROCESS :     3  X  3  X  3.1 4 1 6  =  2  8.2  7  4  4 ,  sq.  in.  in  surface. 

Art,  158.  EuLES.— 1.  To  find  the  area  of  a  rectangle, 
Multiply  the  length  by  the  width. 

2.  To  find  the  area  of  a  triangle,  Multiply  the  base 
by  one  half  of  the  altitude. 

3.  To  find  the  circumference  of  a  circle,  Multiply  the 
diameter  by  3.1416. 

4.  To  find  the  area  of  a  circle,  Multiply  the  circum- 
ference by  one  fourth  of  the  diameter. 

5.  To  find  the  number  of  square  feet  in  a  board  not 
exceeding  one  inch  in  thickness,  Multiply  the  length  in 
feet  by  the  width  in  inches ,  and  divide  the  product  by  12. 

Notes. — 1.  If  tlie  board  is  H  inches  thick,  add  i  of  this  surface 
measure;  if  li  inches  thick,  add  i  of  surface  measure. 

2.  Planks,  joists,  sills,  and  other  timber  are  measured  by  multi- 
plying the  nmiiber  of  square  feet  in  one  surface  by  the  thickness  in  incites. 


LESSON    II. 
Solids, 

1.  How  many  cubic  feet   in   a  block  of  stone  4  ft. 
long,  3  ft,,  wide,  and  2  ft.  thick? 

PROCESS.  ^'     y    ,..-     ..-    ^ 

4  X  3  =  12,  cu.  ft.  in  1  ft.  thick.        (fjimi^'^^^ 

Note.  —  For  fuller  explanation  see  iMliliilliliiililillilift^^ 

Complete  Arithmetic,  p.  105.  iiillMiilM^^ 

2.  How  many  cords  of  wood  in  a  pile  40  ft.  long, 
12  ft.  high,  5  ft.  thick? 


212  INTERMEDIATE  ARITHMETIC. 

3.  How  many  cords  of  four- foot  wood  in  a  pile  64 
ft.  long  and  5^  ft.  high?     (Art.  115,  Note  1.) 

4.  How  many  cords  of  three-foot  wood  in  a  pile 
28  ft.  long  and  6  ft.  high? 

5.  How  many  bushels  of  wheat  will  fill  a  bin  6  ft. 
long,  5  ft.  wide,  and  3  ft.  deep? 

Note. — A  bushel  contains  2150J  cubic  inches. 

6.  How  many  cubic  feet  in  a  round  block  of  timber 
5  ft.  long  and  2  ft.  in  diameter? 

PROCESS. 

2  ft.  X  3.1 4 1 6  --  6.2  8  3  2  ft.,  cir.  of  base. 
6.2  8  3  2  X  2  --  4  =  3.1  4  1  6,  8q.  ft.  in  base. 
3.1 4 1 6  X  5  =  1  5.7  08,  cu.  ft.  in  block. 

Note. — The  block  of  timber  is  a  cylinder. 

7.  How  many  cubic  feet  in  a  circular  well  30  ft. 
deep  and  3  ft.  in  diameter? 

8.  A  liquid  gallon  contains  231  cubic  inches:  how 
many  gallons  in  a  barrel  27  inches  long  and  18 
inches  in  diameter  (average)  ? 

9.  How  many  cubic  inches  in  a  globe  10  inches 
in  diameter? 

flOX  10  X  8.1416  =  314.16,  sq.  in.  in  surface. 
process:  < 

1314.16X1  =  528.6,  cu.  in.  in  globe. 

Art.  159.  Rules. — 1.  To  find  the  solid  contents  of 
a  rectangular  solid,  Multiply  the  lengthy  width,  and 
thickness  together. 

2.  To  find  the  solid  contents  of  a  cylinder.  Multiply 
the  area  of  the  base  by  the  altitude. 

3.  To  find  the  solid  contents  of  a  sphere.  Multiply 
the  surface  by  one  third  of  the  radius. 


MULTIPLICATION  TABLE. 


213 


MULTIPLICATION  TABLE. 


IX 
2X 
3X 
4X 
5X 
«Xl 
7X1 
8X1 
9X1 
10X1 
11  XI 
12X1 


1 
2 
8 
4 
5 
6 
7 
8 
9 

10 
11 
12 


1  X 
2X 
3X 
4X 
5X 
6X 
7X 
8X 
9X 

10  X 

11  X 

12  X 


2  = 
2  = 
2  = 
2  = 

2:= 
2:z:r 
2  = 

2  =: 

2=r 

2  = 
2  = 


IX 
2X 
3X 
4X 
5X 
6X 
7X 
8X 
9X 

10  X 

11  X 

12  X 


3  = 
^^ 
3  = 
3  = 

3=r= 
3=: 
3=r 

3  = 

3   =: 


3 
6 
9 
12 
15 
18 
21 
24 
27 
30 
33 
36 


IX 
2X 
3X 
4X 
5X 
6X 
7X 
8X 
9X 

10  X 

11  X 

12  X 


12 
16 
20 
24 
28 
32 
36 
40 
44 
48 


1X5: 
2X5: 
3X5t: 
4X5: 
5X5: 
6X5: 
7X5: 
"X5: 
X5: 


10  X 

11  X 

12  X- 


5 

10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
60 


IX 
2X 
3X 
4X 
5X 
GX 
7X 
8X 
9X 

10  X 

11  X 

12  X 


6 
12 

18 
24 
30 
86 
42 
48 
54 
60 
66 
72 


IX 
2X 
3X 
4X 
5X 
6X 
7X 
8X 
9X 

10  X 

11  X 

12  X 


7 
14 
21 
28 
35 
42 
49 
56 
63 
70 
77 
84 


IX 
2X 
3X 
4X 
5X 
6X 
7X 
8X 
9X 

10  X 

11  X 

12  X 


16 
24 

32 
40 
48 
56 
64 
72 
80 
88 
96 


X9: 

X9 

X9: 
X9: 
X9: 

X9. 

X9: 

8X9: 

9X9: 

10X9: 

11X9: 

12X9: 


r  9 
-:    18 

--  27 
z  36 
=  45 
3  54 
=  63 
r  72 

r  81 
z:    90 

r  99 
rl08 


1X10: 

2X10: 
3X10: 


4X  10: 
5X10: 
6X10: 
7X10: 
8X10: 
9XJ0: 
10  X  10: 
U    XIO: 

12  X  10: 


10 

1  Xl^ 

[  ^  11 

20 

2X  1^ 

I    :=    22 

80 

3X11 

I   =:    33 

40 

4X11 

L  .=  44 

50 

5X1] 

'--=.  55 

60 

6X11 

I  =  66 

70 

7X11 

\=  77 

80 

8X11 

L  —  88 

90 

9  X  11 

I   =r    99 

00 

10  X  11 

=110 

10 

11  Xli 

[  =121 

20 

12X11 

L  =132 

1    X   12  : 

2X12^ 

3X12: 

4X12: 

5X12: 

6X12: 

7X12: 

8X12: 

9X12: 

10   X   12: 

11X12: 

12  X  12  : 


=  12 
^  24 

r  36 

r    48 

r  60 
=  72 
=  84 
=  96 
=108 

rl20 
nl32 

=144 


ANSWERS 

TO 

THE  WRITTEN  PROBLEMS. 


N.  B.— The  last  answer  is  given  when  a  problem  has  several 
answers,  and  also  when  several  problems  are  united. 


NOTATION. 

Pagre  19. 

15.  3,000,300,000,303.22. 

30,000,075,000. 

10. 

50,032,640. 

16.  62,300,049. 

23. 

9,000,009,009. 

IL 

300,009,206. 

17.  500,005,000. 

24. 

54,087,086. 

Page  20. 

18.  406,507. 

25. 

202,580. 

12. 

48,000,017,064. 

19.  2,010,080. 

26. 

50,050,500,007. 

13. 

5,005,005. 

20.  90,007,490. 

27. 

17,000,700,306. 

14. 

1,100,010. 

21.  400,040,404. 
ADDITION. 

28. 

90,010,055. 

Page  24. 

18.  8,996  pounds. 

6. 

395,096. 

1. 

10. 

Page  26. 

7. 

53,293,685. 

2. 

96. 

1.  280. 

Page  30. 

3. 

899. 

2.  2,^18. 

1. 

197,251. 

4. 

9,889. 

3.  270,724. 

2. 

300,334. 

5. 

87,978. 

4.  270,527. 

3. 

319,076. 

6. 

2,051. 

5.  307,352. 

4. 

40,805,643. 

7. 

174. 

6.  4,444,844. 

5. 

21,316,368. 

8. 

1,264. 

Page  27. 

6. 

51,147,320. 

9. 

14,736. 

7.  15,084  bushels. 

7. 

181  days. 

10. 

861,566. 

8.  1,949  voters. 

8. 

1,462  acres. 

Page  25. 

9.  649  acres. 

9. 

1,388  papers. 

11. 

2,925. 

10.  295,612  pounds. 

10. 

12,326  youth. 

12. 

14,333. 

Page  28. 

Page  31. 

13. 

599,939. 

1.  29,537. 

1. 

4,005. 

14. 

20,195. 

2.  214,353. 

2. 

47,169. 

16. 

95  pounds. 

3.  262,105. 

3. 

347,575. 

16. 

184  days. 

4.  243,571. 

4. 

327;008. 

17. 

787  acres. 
(214) 

5.  1,868,776. 

5. 

46,833. 

ANSWERS. 


Page  32. 

2. 

29,348. 

1. 

102,924. 

6. 

821,412. 

3. 

77,055. 

Page  35. 

7. 

826,826,826. 

4. 

1,374. 

2. 

$10,575. 

8. 

823  miles. 

5. 

25,914. 

3. 

1,552  pages. 

9. 

6,595  houses. 

6. 

10,000,800,999. 

4. 

185,425  sq.  miles. 

10. 

1,064  miles. 

7. 

65,038  sq.  miles. 

5. 

653  miles. 

Page  33. 

Page  34. 

6. 

$14,750. 

1. 

124,112. 

8. 

1,037  miles. 

7. 

1,424  bushels. 

SUBTRACTION. 

Page  38. 

7. 

909. 

Page  44. 

16. 

2,034. 

8. 

404. 

1. 

36,944. 

17. 

354  pounds. 

9. 

7,061. 

2. 

2,633,755. 

18. 

24. 

10. 

3,314. 

3. 

11,971  sq.  miles. 

Page  39. 

11. 

9,825. 

4. 

8,033,009. 

1. 
2. 
3. 

121. 

1,002. 

103. 

12. 
13. 
14. 
15. 

Page  42. 

8,028. 
$485. 

478  youth. 
84,622  pupils. 

5. 
6. 

P^ge  45. 

22,853  miles. 
22,772  men. 

4. 
5. 

132. 
1,141. 

7. 
8. 

14,805,623  lbs. 
$10,337. 

6. 

3,311. 

Page  47. 

7. 

2,026. 

Page  43. 

1. 

42  yards. 

8. 

203  acres. 

1. 

799,762. 

2. 

21,224  sq.  miles. 

9. 

$2,130. 

2. 

39,920,000. 

3. 

599  men. 

10. 

2,062  bushels. 

3. 

277. 

4. 

1,185  bushels. 

11. 
12. 

Page  40. 

$3,235. 

100  sch'l-houses, 

4. 
5. 
6. 

■    7. 

156  years. 
59  years. 
5,618  feet. 
1,511  sheep. 
6,926  feet. 

5. 
6. 

7. 

$240. 
273. 

Page  48. 

745,452. 

13. 

11,115  fleeces. 

8. 

8. 

$6,200. 

Page  41. 

9. 

128  years. 

9. 

3,728. 

6. 

2,297. 

10. 

1,541,000. 

10. 

680,134. 

MULTIPLIOATION 

Page  49. 

6. 

909,609. 

11. 

1,280  rods. 

2. 

9,636. 

7. 

690,963. 

Page  51. 

3. 

30,606. 

8. 

640,402. 

2. 

3,624. 

4. 

69,963. 

9. 

$880. 

3. 

15,144. 

5. 

808,488. 

10. 

$399. 
215 

4. 

31,815. 

ANSWERS. 


5.  28,360. 

6.  54,180. 

7.  151,470. 

8.  1,620  pins. 

9.  3,400  miles. 

10.  945  tons. 

11.  $912. 

12.  $14,180. 

13.  10,080  min. 

14.  124,160. 

Pagre  53. 

2.  32,724. 

3.  37,654. 

4.  56,158. 

5.  338,364. 

6.  1,674,156. 

7.  310,612. 

8.  5,800,509. 

9.  8,208  miles. 

10.  $4,455. 

11.  18,262  yards. 

Pasr^  54. 

2.  16,376,688. 

3.  130,721,463. 

4.  7,188,372. 

5.  8,811,712. 

6.  61,926,668. 

7.  778,822,521. 


8.  330,325  days. 

9.  98,800  pounds. 

10.  2,948,190  lbs. 

11.  $63,500. 

12.  $5,305,125. 

Paire  56. 

2.  1,728,000. 

3.  124,740,000. 

4.  12,685,000. 

5.  1,054,970,000. 

6.  4,250,400  feet. 

7.  3,264,000  miles. 

8.  21,000  times. 

9.  68,600  pounds. 

10.  289,920  sheets. 

11.  94,240  pounds. 

12.  40,600. 

Pagre  57. 

2.  150,400,000. 

3.  33,596,000,000. 

4.  893,000,000. 

5.  342,000,000,000. 

Paee  58. 

6.  2,268,000  sec. 

7.  691,200,000. 

8.  11,700  lbs. 

9.  $425,000. 
10.  36,180  men. 


11.  26,000  miles. 

12.  58,800  stalks. 

PiHpe  59. 

1.  181,078. 

2.  1,931,200. 

3.  18,300,672. 

4.  3,379,200,000. 

5.  3,120,000,000. 

6.  364,450  lbs. 

7.  357,700  lbs. 

8.  268,800  lbs. 

9.  21,312  cents,  or 

$213.12 
10.  473,040. 

Page  63. 

1.  73,696. 

2.  $825. 

3.  $684. 

4.  $288. 

5.  2,700. 

6.  157,248  words. 

7.  $420. 

8.  $18,160. 

9.  $2,400. 

Page  63. 

10.  $1,330. 

11.  $85. 

12.  $80,000. 

13.  25,920  soldiers. 


Pase  64. 

2.  241. 

3.  4,321. 

4.  2,312. 

5.  3,123. 

6.  1,232. 

7.  4,042. 

8.  2,103. 


DIVISION. 

9.  213. 

10.  132. 

11.  180. 

Paee  66. 

2.  191. 

3.  234. 

4.  128. 

216 


5.  174. 

6.  153. 

7.  282. 

8.  1,748. 

9.  3,696. 

10.  93  boxes. 

11.  243  barrels. 

12.  390  days. 


ANSWERS. 


«8-M  I 


Paee  68. 

2.  312. 

3.  312. 

4.  222. 

5.  321. 

6.  527. 

7.  6,546. 

8.  4,725. 

9.  3,436. 
10.  4,326. 

12.  22. 

13.  25. 

14.  48. 

15.  97. 

16.  22  yards. 

17.  436  bushels. 

Page  69. 

18.  76  hogsheads. 

19.  243  boxes. 

20.  47  farms. 

21.  49  days. 

22.  247  years. 

Page  70. 

2.  579,  with  360  R. 

3.  11,  with  17,211  R. 


4.  768,  with  499  R. 

5.  506  weeks. 

6.  405  cattle. 

7.  207  days. 

8.  603  days. 

9.  48  hours. 

10.  41cd.32sq.  ft.R. 

11.  81,073. 

Page  72. 

3.  456. 

4.  1,870. 

5.  384,  witli  50  R. 

6.  23,  with  45  R. 

7.  450,  with  860  R. 

Page  73. 

9.  8. 

10.  40,030. 

11.  937,w'h63,000R. 

13.  234,  with  385  R. 

14.  12. 

15.  64  barrels. 

16.  270  reams. 

17.  48  hours. 

18.  16  h)ts. 

19.  44  cars. 


20.  46  barrels. 

21.  40  regiments. 

22.  109  acres. 

Page  74. 

23.  36  hours. 

24.  156. 

25.  84. 

26.  376. 

Page  78. 

1.  406. 

2.  42,909. 

3.  22,962. 

4.  36. 

5.  82,532. 

6.  48. 

7.  $14. 

8.  $285. 

9.  10  cows. 

10.  $2,010. 

11.  $625. 

12.  3,700  bushels. 

13.  $9,850. 

Page  79. 

14.  60  miles. 

15.  732  miles. 


PROPERTIES  OF  NUMBERS, 


Page  82. 

13.  2,  5, 

5, 

5. 

26.  32. 

2.  3,  3,  7. 

14.  2,  2, 

2, 

5, 

11. 

Page  84. 

3.  2,  2,  2,  3,  3. 

15.  2,  2, 

5, 

5, 

5. 

2.  60. 

4.  2,  2,  3,  7. 

16.  2,  2, 

2, 

3, 

3,  3,  3. 

3.  168. 

5.  2,  2,  2,  2,  2,  3. 

17.  2,  2, 

3, 

3, 

5,  5. 

4.  480. 

6.  5,  5,  7. 

19.  18. 

5.  108. 

7.  3,  7,  7. 

20.  24. 

6.  240. 

8.  5,  5,  11. 

21.  15. 

7.  144. 

9.  5,  5,  13. 

22.  12. 

8.  1,440. 

10.  2,  2,  2,  3,  11. 

23.  27. 

9.  300. 

11.  2,  2,  2,  5,  5. 

24.  21. 

10,  500. 

12.  2,2,2,2,2,2,2,2. 

25.  48. 

11.  144. 

217 


89-103 


Page  89. 

14.  ip. 

15.  HK 

Page  90. 

16.  *F. 

17.  m^. 

18.  -W- 

19.  HiK 

20.  W- 

21.  -W- 

22.  -t-¥-*. 

23.  W- 

24.  mK 

25.  HF- 

26.  Wi- 

97       1860£ 
<»«•    ~2  0(y    • 

Page  91. 

12.  17xV 

13.  6xV 

14.  8tV 

15.  15,V 

16.  6. 

17.  3fr 

18.  lOM. 

19.  2. 

20.  11. 

21.  20,V 

22.  20Jt. 

23.  10. 

24.  mi 

25.  2I3V 

26.  9. 

27.  28ii. 

28.  I7ii 

Page  93. 

14.  |. 
16.  J. 


ANSWERS. 

FEAOTIONS. 

16.  f. 

22. 

21!- 

17.  A- 

23. 

m- 

18.  |. 

19.  i. 

20.  If 

21.  f. 

Paare98. 

25. 

175|. 

26. 
27. 

115|. 
315. 

22.  f. 

23.  tV 

28. 

SolA- 

24.  f. 

Pace  9». 

25.  I5V 

11. 

i. 

26.  t't- 

12. 

h 

27.  h 

14. 

A- 

28.  i 

15. 

h 

29.  f. 

16. 

i«- 

17. 

A- 

Page  95. 

18. 

if- 

12.  If. 

19. 

Ih- 

13.  II,  If. 

21. 

21A- 

14.  H,  ff,  n- 

16.  t\,  f?,  t's- 

17.  if,  if,  if. 

18.  U,  fJ,  il- 

22. 
23. 

24. 

155A. 
66|. 

19.  if,  il.  A- 

Page  100. 

20.  !f.  A,  A- 

9. 

n- 

21.  A,  M,  A- 

10. 

f- 

22.  if.  A,  A,  U- 

11. 

A- 

23.  if,  11,  fl,  If. 

12. 

74iJ. 

24.  A^A^./A.AV 

Pace  101. 

Paee»7. 

13. 

2f. 

12.  21 

14. 

41J. 

*"»•         2* 

13.  2ii. 
15.  lA- 

15. 
16. 

66|. 

16.  2f. 

Page  lOS. 

17.  lA- 

12. 

31. 

18.  m. 

13. 

8f. 

19.  lA- 

14. 

15|. 

20.  2A. 

15. 

21. 

21.  2  A- 

16. 

3^. 

218 

ANSWERS,  II 

17.  in.  22.  |.  16.  32. 

18.  n.  23.  /s-  17.  12. 

19.  66|.  24.  i  18.3. 

20-  225.  25.  f.  .            Pai^  .,«. 

raee  103.  26.  f.  2.  IJ. 

13.  305|.  11  \\  3.  IJ. 

14.  loij.  ;°-  '•  4.  f. 

15.  2,112^.  30.,.  5.1^. 

16.  l,972i.  il-  f  6.  I. 

17. 192.  ^i-  ^;  7. 2|. 

<><*•  Ti-  8.  3|. 

Page  104.  34.    1.  g    g| 

18.  20.  35.  f.  lo!  A. 

19.  56|.  36.  J. 

20.  50f.  37.  If  . 

21.  69|.  38.  6}.  ^ 

22.  USA.  39.  IH.  J  5-         ,„,, 

23.  260.  40.  15|.  ^-  f  "»'  T^s- 

24.  176H.  41.781.  *-;|„, 
25.2561.                              r^,^  ^'''^"■ 

26.  119,V.  1,     f"*****-  rareiw. 

27.  96f.  "•  V  6.  llH. 

28.  247if.  f  ;•  Tf  7. 


Pace  111. 


T¥- 


29.  553H.  ;  J  \^^-                         8.  5,536. 

31. 792.  ;:•  \^-                 9.  A- 

32.  1,350.  !^-  V  10.  46i-V 

33.  2,292.  }°-  Y-  11.  ItV 

34.  3,136|.  ,  ^-  Y'  12.  1,188  feet. 

35.  5,670.  ■  iQ    f;  13.  $3,750. 

36.  13,5331.  g^-  7/-  14.  A. 

37.  6,094f.  i^Z^f  15.  Soldi;  owns  f. 

38.  38,042|.  i^-  gf  16.  1,089  feet. 

39.  45,600.  **•     ^"  17.  12  rods. 

Pace  100.  18.  24  barrels. 

Page  106.  jQ_  24.  19.  $21,000. 

17.  f.  11.  20.  20.  $8,400. 

18.  h  12.  75.  21.  2,016  bushels. 

19.  h.  13,  68f  22.  $10,560. 

20.  /j.  14.  49.                                          Paite  113. 
21'  A-  15.  70.  23.  $6,832. 

219 


ANSWERS. 


Paice  120. 

6.  .4500 

7.  6.500 

8.  23.00 

9.  62.5000 

10.  .04800 

11.  406.062000 

13.  .5 

14.  2.40 

Page  121. 


1 
J' 

3 


4.  }. 

5.  |. 
6. 
7. 
8. 
9. 

10. 
11. 

12.  3 J. 

13.  21^3^. 

14.  ^1,. 

15.  12|. 

16.  25^1^. 

4.  .75 

5.  .625 

6.  .0625 

7.  .125 

8.  .075 

9.  .024 

10.  .256 

11.  .0075 

12.  3.08i 


DECIMAL  rRAOTIONS. 

13.  21.75 

14.  .0032 

15.  12.0375 

16.  25.032 

17.  .625 

18.  .0124 

19.  12.062500 

Page  122. 

5.  28.2104 

6.  182.097 

7.  $926,498 

8.  5378 

9.  178.455  miles. 

Page  123. 

6.  16.544 

7.  .032625 

8.  .011992 

9.  .7992 

10.  31.48 

11.  .0092 

Page  125. 

12.  .25331 

13.  .1728 

14.  .4355 

15.  .18468 

16.  161.5 

17.  1505.52 

18.  156. 

19.  1.3332 

20.  .000056 

21.  .03625 

22.  121. 


23.  .021 

24.  3.1828 

25.  .00156 

27.  4085. 

28.  300480. 

Page  127. 

9.  3.6 

10.  .28 

11.  11.2 

12.  .79 

13.  112 

14.  8.5 

15.  .8 

17.  160. 

18.  200. 

19.  1500. 

20.  230000. 

21.  23000. 

23.  .4367 

24.  .002346 

Page  12!^. 

1.  .001 

2.  .0064 

3.  iin?. 

4.  8.7 

5.  1020. 

6.  1120. 

7.  .1743 

8.  .0072 

9.  176000. 
10.  $721,446. 


UNITED  STATES  MONEY. 


Page  132. 

8.  560  c. 

11.  3125  m. 

6.  10800  c. 

9.  801  c. 

12.  105  m. 

7.  23000  c. 

10.  40000  ra. 
220 

13.  3030  m. 

ANSWERS. 


14.  50  m. 

15.  1000  c. 

16.  45000  m. 

17.  ic. 

18.  $75. 

19.  $75.50 

20.  $3,125 

21.  $4. 

22.  $15.07 

23.  $10.01 

24.  $1.50 

25.  $10.25 

26.  $50. 

27.  $5. 

28.  $3.75 

29.  $.375 

Page  133. 

30.  4500  m. 

31.  $4.50 

32.  1010  c. 

33.  $1.01 

2.  $17,825 

Page  134. 

3.  $97,285 

4.  $308,365 

6.  $.49| 

7.  $98,744 

8.  $9.90 

9.  $327.75 

10.  $1,375 

11.  $2.50 

12.  $4,875 

13.  $493.25 


14.  $2.75 

15.  $281.73 

Page  133. 

2.  $120. 

3.  $45. 

4.  $144.90 

5.  $2.25. 

6.  $87.50 

7.  $1200. 

8.  $146.25 

10.  $.18 

11.  $37.50 

Page  136. 

12.  $7.50 

13.  $11,375 

15.  60  bushels. 

16.  200  lemons. 

17.  32  days. 

18.  154  bushels. 

19.  56  yards. 

20.  80  sheep. 

21.  40  lemons. 

22.  75. 

23.  36. 

Page  137. 

24.  250. 

1.  $139.59 

2.  $9,995 

3.  $495. 

4.  $506. 

5.  100. 

6.  $131.60 


7.  $600. 

8.  $20. 

9.  $277.20 

10.  200  yards. 

11.  400  bushels. 

Page  138. 

12.  $1600. 

13.  $1.75 

14.  50  acres. 

15.  $6.75 

16.  $58.50 

17.  $1950. 

18.  $191.25  (365 

days  in  year). 

19.  $405. 

20.  98  yards. 

21.  40  sheep. 

22.  $2.50 

23.  $1230. 

Page  139. 

24.  $116.95 

25.  $71. 

26.  $215. 

Page  140. 

2.  $140.15 

3.  $91.79 

Page  141. 

4.  $132.95 

5.  $288,375 

Page  143. 

6.  $329. 

7.  $26,125 


DENOMINATE  NUMBEES. 


Page  146. 

21.  768  pt. 

Page  147. 
23.  366  pt. 


24.  971  pt. 

25.  280  qt. 

26.  59  pt. 

27.  31  qt. 

221 


28.  66  pt. 

30.  5  bu.  1  pk. 

31.  5bu.  1  pk.  3qt. 

32.  3  pk.  1  qt.  1  pt. 


147-166 


ANSWERS. 


33.  2  pk.  2  qt.  1  pt. 

34.  4  bu.  2  pk.  7  qt. 

35.  $2.90 

36.  14.88 

37.  $4.25 

38.  9bu.  2pk.  6qt. 

39.  112  baskets. 

40.  $1.00 

Page  149. 

17.  168  pt. 

18.  249  gi. 

19.  273  pt. 

Paere  150. 

20.  77  pt. 

21.  4gal.  3qt. 

22.  7  gal.  2  qt.  1  pt. 

23.  16qt.  2gi. 

24.  17  gal.  3  gi. 

25.  1456  gi. 

26.  449  pt. 

27.  38  gal.  1  pt. 

28.  $41.60 

29.  16  vials. 

30.  21  jugs. 

31.  $10. 

32.  $36.60 

33.  $48.30 

34.  $56.70 

Page  153. 

21.  576  in. 

22.  27828  in. 

23.  1819  ft. 

24.  6490  yd. 

25.  5mi.lfur.10rd. 

26.  4fur.  2rd.  5  yd. 

1  ft.  6  in. 

27.  258820  in. 

28.  4224  steps. 


29.  2200. 

30.  12  mi.  3  fur.  3rd. 

3  yd.  1ft.  6  in. 

31.  240  rd. 

Page  156. 

15.  627264  sq.  in. 

16.  880  P. 

17.  70882  sq.  ft. 

18.  1  A.  2  R.  20  P.  10 

sq.  yd.  7  sq.  ft. 

19.  33  A. 

20.  4  sq.  yd. 

21.  14  A. 

22.  20  A. 

23.  324  sq.  yd. 

24.  $125. 

25.  480  trees. 

26.  10000  shingles. 


Page  157. 

16000  A. 
26880  A. 
41 J  yd. 
3442i  bricks. 


27. 
28. 
29. 
30. 

31.  $37. 

Page  158. 

1.  55296  cu.  in. 

2.  9  cu.  ft. 

3.  3240  cu.  ft. 

4.  15  cu.  yd. 

5.  728793  cu.  in. 

6.  31  cu.  yd.  15  cu. 

ft.  1206  cu.  in. 

8.  1872  cu.  ft. 

9.  576  cu.  ft. 

10.  IO3L  cu.  yd. 

11.  220cu.  yd. 

Page  159. 

12.  2160  cu.  yd. 

13.  $406.25 

21i2 


Page  160. 

1.  5  cd.  ft. 

2.  96  cu.  ft. 

3.  44  cd.  ft. 

4.  16  cd. 
6.  12  cd. 

6.  3  cd.  96  cu.  ft. 

7.  5  cd.  20  cu.  ft. 

8.  $49.50 

9.  41  cd.  32  cu.  ft. 
10.  $31,993^ 

Page  161. 

6.  55800"^ 

7.  933^ 

Page  162. 

8.  9900'. 

9.  3°  or  iV  S. 

10.  90°. 

Page  164. 

11.  54000  sec. 

12.  8h. 

13.  481200  sec. 

14.  2680245  sec. 

15.  21  d.  6  h.,  or 

3  w.  6  h. 

16.  6  c.  yr. 

17.  527040  min. 

18.  31556928  sec. 

19.  31536000  sec. 

20.  562116  h. 

21.  2208  h. 

22.  40320  min. 

23.  3780000. 

24.  13  days. 

Page  166. 
9.  160000  oz. 

10.  7456  lb. 

11.  9245  oz. 

12.  17T.9cwt.20Ib. 


ANSWERS. 


166-181 


13.  2cwt.  851b. 

14.  5641968  dr. 

15.  $11.76 

16.  $90. 

Page  167. 

17.  20bbl. 

18.  $105. 

19.  $54. 

20.  $507,875 

21.  61  lb. 

Page  168. 

7.  10633  pwt. 

8.  3720  gr. 

9.  322872  gr. 

10.  56  lb.  2  oz.  6  pwt. 

11.  7  lb.  3  pwt.  16  gr. 

12.  473Jlb. 

13.  11  oz. 

14.  $3150. 

15.  $52.50 

16.  60  rings. 

Page  170. 

7.  97790  gr. 


8.  4980  gr. 

9.  3  lb.  8  g  4  5. 

10.  41b.7gl3l9 

4gr. 

11.  20  1b. 

12.  18  doses,  with 

16  gr.  R. 

13.  128  pills. 

14.  6g. 

Page  171. 

6.  6000  sheets. 

7.  3252  sheets. 

8.  $127.50 

9.  5184  crayons. 

10.  288  shirts. 

11.  $396. 

12.  $22.50 

Page  175. 

15.  $22.50 

16.  $77.50 

17.  $17600. 

Page  176. 

18.  960  times. 


19.  126720  times. 

20.  23040  acres. 

21.  $600. 

22.  24200  pills. 

23.  13333  p'ple,  with 

tV  sq.  yd.  R. 

24.  $10642.50 

25.  120  sq.  yd. 

26.  30|  yd. 

27.  14400  shingles. 

28.  128  rods. 

29.  336  cu.  ft. 

30.  48  P. 

31.  $QQ. 

32.  $40.50 

33.  2592000  times. 

Page  177. 

34.  7305  hours. 

35.  80  rings. 

36.  320  1b. 

37.  120  gross. 

38.  41f  reams. 


COMPOUND  NTJMBEES. 


Page  1*8. 

2.  89  bn.  3  pk.  5  qt. 

3.  121  gal.  3  qt.  1  pt. 

4.  99  mi.  35  rd.  2  ft.  11  in. 

5.  60  cwt.  77  lb.  2  oz.  13  dr. 

6.  55  lb.  3  oz.  8  pwt.  7  gr. 

7.  113  1b.73  25  29l5gr. 

8.  29  w.  6  d.  10  h.  41  min.  29 

sec. 

9.  7  S.  8°  43^  S'^ 

10.  36  sq.  mi.  134  A.  30  P. 

11.  153  cd.  1  cd.  ft  9cu.  ft. 

12.  23  bundles,  10  quires. 

Page  179. 

13.  11  cwt.  61  lb.  10  oz. 


14. 
15. 

2. 

3. 
4. 
5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


223 


18rd.  3  yd.  1  ft.  7  in. 
161  gal.  2  qt. 
Page  181. 

19  cwt.  25  1b.  12  oz.  6  dr. 

3  1b.  11  g  1  .5  1  9  18  gr. 

4  w.  1  d.  20  h.  12  m. 

20  mi.  6  fur.  27  rd.  4  yd.  1 
ft.  6  in. 

8  rd.  1  yd.  11  in. 

17  gal.  3  qt.  1  pt.  3  gi. 

2  bu.  5  qt.  1  pt. 

207  A.  2  R.  29  P. 

45°  23^  27"^ 

1  T.  12  cwt.  35  lb. 

24  gal.  1  pt. 


181-18(1  ANSWERS. 

13.  2  lb.  3  oz.  4  pwt.  10  gr.  9,  19  cwt.  98  lb. 

14.  37  mi.  4  fur.  20  rd.  10.  14  pwt.  14  gr. 

Page  182.  1 1.  57  A.  2  E.  24  p. 

16.  9  yr.  7  mo.  22  d.  12.  43  gal.  2  qt.  If  pt. 

17.  4  yr.  1  mo.  10  d.  13.  4  rd.  3  yd.  1  ft. 

18.  56  yr.  2  mo.  3  d.  14.  2  sq.  rd.  16  sq.  yd.  2  sq.  ft. 

19.  7  yr.  9  mo.  Id.  15.  3°  13^  39^^ 

20.  283  yr.  8  mo.  20  d.  17.  52  bottles.      18.  4  baskets. 

21.  2  yr.  9  mo.  12  d.  Page  187. 

22.  March  15,  1767.  19.  45  lengths. 

Page  183.  20.  44  castings. 

2.  124  lb.  5  oz.  4  pwt.  21.  1131  f  times. 

3.  95  yd.  2  ft.  22.  70  rings,  with  80  gr.  R. 

4.  92  mi.  5  fur.  12  rd.  23.  1697^  steps.      24.  8  hours. 

Page  184.  Page  188. 

5.  35  lb.  6  g  2  3  2  9.  1.  5  lb.  11  oz.  IH  dr. 

6.  251  bii.  5  qt.  2.  3  lb.  1  oz.  4  pwt. 

7.  34w.  3d.  19  h.  3.  2450  gal. 

8.  14  cwt.  73  lb.  4.  19  lb.  9  oz.  16  pwt. 

9.  4  oz.  11  pwt.  12  gr.  5.  32  yr.  4  mo.  18  d. 

10.  270  h.  6.  13  gal. 

1 1.  51°  23^  15^^  7.  78  A.  1  R.  32  P. 

12.  2571  yd.  8.  5  T.  12  cwt.  50  lb. 

13.  607  bu.  3  pk.  4  qt.  9.  2  cwt.  54  lb.  4  oz. 

14.  38  yr.  11  mo.  20  d.  10.  2  reams,  5  quires. 

15.  12  cd.  5  cd.  ft.  4  cu.  ft.  11.  43  cd.  4  cd.  ft.  4  cu.  ft. 

16.  149  gal.  2  qt.  Page  189. 

17.  29  T.  20  lb.  12.  2  T.  17  cwt.  10  lb. 

18.  2  fur.  18  rd.  3  yd.  2  ft.  13.  1  bu.  1  pk.  2i  qt. 

19.  90  reams,  12  quires.  14.  12  bu.  3  pk.  3f  qt. 

20.  35  mi.  5  fur.  24  rd.  15.  1188  steps. 

21.  143bu.  3pk.  16.  4658}^  times. 

Page  186.  17.  42  yd.      18.  82|f  yd. 

2.  1  lb.  13  oz.  9 J  dr.  r  A.  6  cwt.  80  lb. 

3.  5  yd.  1  ft.  2  in.  19.    <   B.  4  cwt.  53i  lb. 

4.  10  A.  2  R.  14  P.  i  C.  2  cwt.  26f  lb. 

5.  1  cwt.  62  lb.  4  oz.  lOf  dr.                    r  1st  mo.,  8  m.  4  fur.  25|  r. 

6.  5w.6d.llh.5min.l6T\sec.      q^     j  2d     "     6m.3fur.  19r. 

7.  2  lb.  6  5  1  9  14f  gr.  **    •    I  3d     "     9  m.  5  fur.  8i  r. 

8.  7  h.  30  min.  i  4th  "     1  m.  0  fur.  23^  r. 

224 


ANSWERS. 


192-21S 


PEEOENTAGE. 


Page  192. 

7.  $260. 

8.  14.7 

9.  108. 

10.  96.6 

11.  77. 

12.  105  ft. 

13.  184.8  1b. 

14.  10.7  oz. 

15.  120  men. 

16.  $2.25 

17.  $3.15 

18.  2.43ida. 

19.  $28.95 

20.  $.03| 

21.  $.516 

22.  1.215  ft. 

23.  6.118  mi. 

24.  $1587.50 

25.  190  acres. 

26.  392  sheep. 

27.  $13430. 
Pagre  193. 

28.  1859  men. 

29.  264  pupils. 

30.  1029  b.wH. 
588  b.  oats. 
833  b.  corn. 

Pagre  194. 

10,  18/,. 

11.  7if,. 


12.  5f,. 

13.  14/,. 

14.  12J/,. 

15.  4iy,. 

16.  15/,. 

17.  1/.. 

18.  no/,. 

19.  10/,. 

20.  105/, . 

21.  85/,. 

22.  40/, . 

23.  80/,. 

24.  16f /, . 

25.  40/,. 

26.  66f/,. 

27.  88/,. 

28.  8/,. 

29.  70/, . 

30.  24/,. 

Page  195. 

10.  200. 

11.  $462. 

12.  $56. 

13.  $65.28 

14.  $216. 

15.  $70. 

Page  196. 

16.  28001b. 

17.  $54000. 

18.  $2200. 


Page  197. 

15.  $12250. 

17.  $21692. 

18.  $175.50 

19.  71-/,. 

20.  $1.08 
$1.26 
$1.32 

Page  198. 

1.  $175. 

2.  $32.45 

3.  $635. 

4.  $93.60 

6.  $41. 

7.  $18.75 

8.  $94. 

Page  199. 

10.  $62. 

11.  15  mills. 

12.  $12225. 
Page  201. 

18.  $.106 

19.  $.129i 

20.  $.198i 

21.  $.069 

22.  $.130 J 

23.  $.065 J 

24.  $.036i 

25.  $.300f 
27.  $19,146  + 


Page  202. 

28.  $1,765  + 

29.  $106,162  4- 

30.  $207,746 

32.  $39.726| 

33.  $154,837  + 

34.  $524.40 

35.  $303,062 

36. 


Page  204. 

2.  $234,384  + 
Page  205. 

3.  $22,145  + 

4.  $714.72 

5.  $61,568 

6.  $122,578  + 

8.  $138,686  + 

9.  $354.88 
10.  $224,347  + 

Page  206. 

12.  $142.50 

14.  $11750. 

15.  $1425. 

Page  208. 

2.  $2468.375 

3.  $5565. 

6.  $1962.80      ■ 

7.  ^175250. 
9.  $5.62i 

10.  $550. 


Page  200. 

2.  9  acres. 

4.  12  sq.  yd. 

5.  7  sq.  ft. 


MENSUEATION. 


Page  210. 

6.  301  sq.  ft. 

7.  480  sq.  ft. 
9.  47.124  ft. 


11.  5026.56  s.  ft. 

Pp.  211,  212. 

2.  18|  cords. 

3.  11  cords. 


225 


72.32 +  bu. 
212.058  c.f. 


8.  29.743  +  g. 


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THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


ECLECTIC  EDUCATIONAL  SERIES. 

Tew  Eclectic  Penmanship. 


n^HE  simplest,  7?iost  leg^ible,  and  business-like  style  of  Capitals  and  Small 
Letters  is  adopted.  In  the  Copy-Books  each  letter  is  given  separately  at 
Jlrst,  and  then  in  combination ;  the  spacing  is  open ;  analysis  simple,  and 
indicated  in  every  letter  when  first  presented ;  explanations  clear,  concise, 
and  complete  are  given  on  the  covers  of  the  books,  and  not  over  and  around 
the  copies. 

NEW  ECLECTIC  COPY-BOOKS.— Revised  and  Re-engraved. 

No's  I,  2,  3,  4,  5  Boys,  5  Girls,  6  Boys,  6  Girls,  6 3^,  7,  8  Boys,  8  Girls,  and 
No.  9.  Girls'  Copy-Books  identical,  word  for  word,  with  the  Boys' ,  but 
in  smaller  hand-writing.     First-class  paper,  engraving,  and  ruling. 

ECLECTIC  ELEMENTARY  COURSE 

The  Elementary  course  comprises  three  books,  smaller  than  the  Copy 
Books,  but  the  same  in  form.     No's  i  and  2  are  Tracing-Books. 

ECLECTIC  PRIMARY  COPY-BOOK. 

A  complete  Primaiy  Penmanship,  designed  for  use  during  the  second  year 
of  school  life.  It  contains  all  the  small  letters,  figures  and  capitals,  each 
given  separately  and  of  large  size,  the  object  being  to  teach  the  form  of 
the  letter.  It  is  designed  to  be  written  with  the  lead-pencil.  Furnished 
either  in  white  or  manilla  paper. 

ECLECTIC  EXERCISE-BOOK. 

Contains  a  variety  of  exercises  especially  designed  to  develop  the  differ- 
ent movements,  and  so  arranged  as  to  give  as  much  or  as  little  practice 
on  each  exercise  as  may  be  desired.  It  is  a  little  larger  than  the  Copy- 
Books,  and  has  a  strong  cover,  so  that  the  latter  may  be  placed  within  it, 
thus  making  it  convenient  to  keep  the  two  together. 

THE  ECLECTIC  PRACTICE-BOOK 

Is  made  of  the  same  size  and  weight  of  paper  as  the  Copy-Books,  ruled  with 
-'ouble  Hues  for  No's  i,  2,3,  4,  and  with  ?^ngle  lines  for  the  higher  number-.. 
NEW  HAND-BOOK  OF  ECLECTIC  PENMANSHIP. 

A  Key  to  the  Eclectic  r  ^steni  oi  Penmanship.  A  complete  description 
and  analysis  of  movement  ana  ■  the  letters,  and  a  brief  summary  of  what 
is  required  in  teaching  penmauonip, 

ECLECTIC  WRITING-CARDS. 

72  No's  on  36  Cards.      Oue  T.etter  or  Principle  on  each   Card:    Capital 
Letter  on  one  side.  Small   Letter  on  the  reverse.     Each  illustration  accom- 
panied  with    appropriate    explanations   and    instructions.     Size    of  Cards, 
9x13  inches ;  loop  attached  for  suspending  on  the  walls. 
SAMPLE  BOOK  OF  ECLECTIC  PENMANSHIP. 

Containing  nearly  200  copies  selected  from  all  the  Copy-Books  in  the 
Series.  Will  be  sent  by  mail  for  15  c.  to  any  teacher  or  school  officer  desir- 
ing to  examine  it  with  a  view  to  introducing  the  Eclectic  Penmanship. 

ECLECTIC  PENS. 

School  Pen,  No.  100,  90  c.  per  gross;  small  box  (2  doz.),  25  c.  Com- 
mercial Pen,  No.  200,  90  c.  per  gross.  Ladies'  Pen,  No.  300,  90  c, 
per  gross.  Free  Writing*  Pen,  No.  400,  90  c  per  gross.  Sample  Card 
of  Eclectic  Pens,  10  c. 


VAN  ANTWERP,  BRAGG  &  CO.,  Publishers,  Cincinnati. 


